Soundtrap is an interesting DAW option for anyone with unlimited internet and has been around for some time now. 
Writing this as they have a specific Android app to use their DAW. Which makes it easier than connecting to their website first, just making everything more streamlined.
There are options out there, the ones that come to mind is Audiotool being more electronic music making friendly and Bandlab being very focused on the social and collaborative aspect (Well to be honest Soundtrap as well).
But as always what generates most inspiration should rule and maybe Soundtrap is better? 
(Have not used any of these three... touched Audiotool lightly and found it to be very interesting with a lot of virtual instruments and effects.)

They have pricing plans and the free one have limits set to five projects and fewer things in general.
In this aspect, Auditool and Bandlab are better for sure.

Google Play:
play.google.com/store/apps/details?id=com.soundtrap.studioapp

Plug in your own instrument, use the software instruments available in Soundtrap or just record a song directly with a mic— it's super-easy!

You can record, edit and collaborate on your songs wherever you are on almost any device: everything is stored in the cloud! 

Moreover, with Soundtrap you don't have to be alone in your creative moments. You can share the fun of making music in real-time with friends using text and video chat and real-time collaboration.

For example, start a new song using all the built in professional loops and software instruments on your tablet when on the bus, continue on your Chromebook when arriving at home, invite a couple of friends, who are working on Mac and Windows devices, and conduct a video chat from within the studio coming up with the bass line for the song and finally find a didgeridoo player in the Soundtrap community laying down a unique ending of the song; you are all working in the same project seeing and hearing all the changes. 

It's like Google Docs, but for music production! Experience the new way of making music together. 

Soundtrap Features:

- Create music with a large set of high-quality, professional loops. 
- Record vocals, electric guitars, acoustic guitar, bass and more. 
- Use the preamp and the large number of high-quality guitar, bass and vocal effects
- Play the built-in sampled instruments (piano, organ, synths, drums)
- Save your recordings in the cloud
- Invite friends to collaborate on the recording using, for example, video chat possibilities. 
- Share your recordings on Facebook, Twitter and SoundCloud
- Distribute your music on iTunes, Deezer, Spotify, Rhapsody and Google Play
- Use it on Windows, Mac, Chromebook, Linux, iOS and Android

Okay this is not Android related but it is important.
Specially for people that does not have any DAW for their computer.
Can not see anything close to this in question of advanced Free music software.
A full blown DAW for free is pretty amazing and it is not like ten years old or whatever,
it is a couple of years old and have everything that can be expected from a modern DAW.

Tried an earlier version and really like the one window approach as everything is there reachable to drag or click up front.
So not much menu diving and how they set up FX chains and automation etc is great.
But personally am so used to Reaper and use that when recording stuff as it is so easy to cut up and move around audio in that DAW.
But if I was composing in a DAW Tracktion would definitively be high on the list and being for free... that would be the one I went for. (Unless SunVox releases a version with audiotracks...)

The only limitation is that there is no technical support and it will not be updated.
But there is a lot of tutorials and there is a forum for help and even if it will never be updated it will work for a long time, would guess as long as it took for Windows XP got discontinued,
could be, guess, like fifteen years or so?

Thank you Tracktion for letting this out in the wild!!!

Website and download:
https://www.tracktion.com/products/t7-daw

HumBeatz are the same developers as a the DAW Amped Studio that is an online DAW that can be used Offline as well allowing for VST/i's.
​When writing about HumBeatz my thought was that it could have been a good idea to have an sequencer buillt in to expand on the two beats allowance but guess that this is a good idea as well and being able to use Amped Studio over the net allows for easy integration when having internet access.

Google Play:
https://play.google.com/store/apps/details?id=com.humbeatz.humbeatz

To read more or try out the DAW (There is a free version and a subscription version for around five moneys a month):
​ampedstudio.com/

I want to thank Andy for bringing this to my attention and it must be good news for anyone using Studio One as their DAW. Being an official app from PreSonus they use the same setup as the DAW and they made it for free, it seems like a perfect controller if using Studio One.

Google Play:
https://play.google.com/store/apps/details?id=com.presonus.dawremote

PreSonus® Studio One® Remote is a free Android remote control app designed specifically for use with PreSonus Digital Audio Workstation Studio One 3 Artist* and Professional on Mac® and Windows® computers. It’s the perfect companion, both as a “second screen” app in a workstation setup or as a flexible mobile remote for recording, mixing and editing when away from the computer.

Studio One Remote is based on the PreSonus software applications framework and utilizes the UCNET protocol for network connectivity and remote control. This is the same technology that powers PreSonus remote control apps such as UC Surface, as well as the popular multitrack live recording software Capture™ (desktop) and Capture for iPad.

Key Features:

- Remote control of Studio One 3 transport and mix console
- Commands page for access to all Studio One factory and user commands and macros
- Control up to 28 plug-in parameters using Control Link
- PreSonus UCNET networking technology for ultra-fast connectivity
- Macro Control view for quick access to FX parameters
- Fast song navigation using scalable timeline, marker list and Arranger sections
- Control any Studio One system on the same network; control a single Studio One with multiple remote apps simultaneously
- Start Page with demo mode and integrated Quick Help
- Access multiple Cue Mixes with independent faders
- Access record modes, Pre-count and Metronome settings
- Remote control over critical PreSonus hardware functions (currently supported products: Studio 192-series USB 3.0 interfaces, Quantum Thunderbolt™ interfaces, StudioLive AI-series mixers, and StudioLive Series III mixers)

REQUIREMENTS:

Compatible with Android devices running Lollipop 5.0 or later and a screen resolution of 1024 x 768 or higher. The Android device must be connected to the same wireless network as a Mac or Windows computer running Studio One 3 Professional.
* Studio One 3 Artist support requires the Studio One Remote add-on. Studio One add-ons are available at shop.presonus.com.

Looking at the images for this one it is almost exactly the same as FL Studio Mobile!
Hopefully it is better though as it seems that a later update have made FL Studio Mobile getting a lot of one star reviews because of not recognizing installments and force closing etc...
(One thing that Music Band does not have is the integration with FL Studio of course.)

It is all sample based instruments and seems to be more to the realistic side of things but have not tried the app so do not know for sure.
There is a free version that allows for testing which is great and would have saved a lot of grief for disappointed buyers of FL Studio Mobile if imageline did not think with greed as a first priority (at least that is how it seems to me...).
I am convinced that it is the same developer/s as for FL Studio Mobile as there is so much similarities... even the same irritating midi note input in the sequencer!
Hopefully it will work more smoothly though.
​Guess I have to test this one day...

In either case there is a discount for a week and there is like said a free version to see if it works okay. Definitively interesting as there is a lot of instruments plus a lot of Audio tracks, recording, effects etc.

The official info:

*** Introductory Sale ***
To celebrate the release, the full version of Music Studio is 33% off for one week!Music Studio offers a complete music production environment for mobile devices with features and a sound quality previously only known to desktop applications and expensive audio hardware.
It combines a piano keyboard, 11 studio-quality instruments (118 in the full version), a fully fledged 127-track sequencer, extensive note editing, reverb, real-time effects and much more on a user-friendly interface.
FAQ
• Why does the Android version have less features than the iOS version?
Music Studio for Android is several versions behind (2.0 vs. 2.6). Both versions will be consolidated in version 3.0. For details, see www.xewton.com or the in-app help.
• Why is there a latency between touching a key and hearing the sound?
The latency is quite high on many Android devices, see the FAQ in the in-app help.
• Black bars are visible at the left and right screen borders on some Android devices.
This will be resolved in version 3.0.
AWARDS & REVIEWS OF THE FULL VERSION
• "A recent composition of mine started on the iPhone with Music Studio" - Jason Hayes, lead composer of "World of Warcraft"
• "Music Studio is loaded to the brim with features and is a no brainer for composers" - AppAdvice.com
• "If you’re at all serious about your music you can’t really afford to miss out on this app." - razorianfly.com
FEATURES
• Photorealistic dynamically configurable 85-key keyboard
• Instant positioning via the slide gesture and resize with the pinch gesture
• 11 studio-recorded instruments (16bit 44.1kHz sampled from real instruments)
• Release and attack time configurable per instrument
• Sustain samples
• Pitch bend wheel and accelerometer control
• Audio tracks:
• Microphone recording
• Audio file import (wav and m4a)
• Waveform and audio region editing
• Low-latency, highly optimized, 128x polyphony, battery saving audio engine
• 10 drum loops
• High-quality reverb and resonant filter and limiter effect
• 127-track sequencer
• Beat & metronome settings (tempo and signature)
• Mute, solo, effect bus, pan and volume adjustment for each track
• Edit tracks, bars and even individual notes:
Draw, quantize, transpose, repeat, move, length, velocity, etc.
• Unlimited undo and redo
• Detailed in-app help
• Play or record 2 different instruments at the same time with 2 keyboard rows
LIMITATIONS OF THE LITE VERSION
• Saving, exporting and sharing is disabled, hence only one song can be created.
• Only 11 instead of 118 instruments are available.
• The number of audio tracks is limited to 2 (up to 127 in the full version).
• 3 Effects are not available: Delay, EQ and Amp
• Only 10 instead of 100 beats
• MIDI import/export is disabled.
• No add-on packages (+60 instruments)
Visit the forum at www.xewton.com - Questions, feedback and bug reports are welcome!
Video tutorials are available on the Xewton YouTube channel.

If using Pure Data I guess this must be a DAW that would be highly interesting!
I asked n-Track Studio if it would be integrated into the Android version of n-track Studio as well and it seems that there is a possibility! 
That would be amazing as there is a lot of instruments and effects made already to be used even if there is no knowledge on how to use Pure Data. 
Songtree is also implemented directly into the DAW which is enticing if you are a user of Songtree on Android.

To read more about this affordable and capable DAW go here:
http://en.ntrack.com/index.php

It is made to work cross platform and of course for Android as well. This  is maybe no surprise as it is cloud based. There is a possibility to use it for free but with the limitation of five projects and some smaller limitations... but it is great if there is a need to collaborate with someone and it is in difference to Audiotool more for recording than midi and sound generators.
So if there is a steady internet connection this is one of the more complete apps there is for Android!
To read more:
https://www.soundtrap.com/

Showing of the latest new function in this mobile Daw that just keeps getting stronger!

Another short review of the Ableton Live controller...
Go here to read it:
http://askaudiomag.com/articles/review-lk-ipad-and-android-ableton-live-midi-controller?utm_content=buffer32cc8&utm_medium=social&utm_source=twitter.com&utm_campaign=buffer

Here is a short review of the Ableton Live controller. It is made using a iPad but the Android version is the same...

To read the review go here:
http://www.residentadvisor.net/review-view.aspx?id=17074

This has been out for awhile and last time I took a look about a year ago I did dismiss it but after accidentally seeing this later basic video it looked much more interesting and can see how this would be perfect for some people.
Maybe the one thing missing would be time stretch.

Play Store info:
Androsynth is a full featured multi-track audio composer, audio editor and effect creator. It contains an audio editor which you can use to import audio files, add effects to them and slice them in samples, an effect editor that allows you to create your own effects from elementary blocks, and a tracker that you can use to compose new music using those samples and effects.Some features are limited in the demo version: songs are limited in duration and there is a maximum number of elements you can use to create effects. There is no limitation in the sampler. 

Sampler features :
► Import most audio file formats including MP3, Vorbis, FLAC, WAV, even audio tracks from videos
► Record audio from the internal microphone
► Add effects such as pitch change, low pass filter, wah-wah, etc ...
► Convenient zooming and editing features
► Save as sample

Tracker features:
► Unlimited number of tracks
► Individual mute/solo/volume control for each track
► Expand and repeat samples
► Add effects and filters independently to each voice
► Master song as WAV in CD quality (44.1kHz, 16bits, stereo) 
► Save/load project
► Zooming, panning, full multitouch support

Effect editor features :
► Create your own audio filter effects
► Start from a preset library of effects that you can modify
► Use your effects anywhere in the application

※ For optimal use it is recommended to use this application with tablets or phones with a 4.7inch or larger screen.


There is more tutorials on the Youtube channel and on the official website:
http://www.draliv.com/androsynth/en/

* Support for 24-bit WAV files
* New loop editing features
* Auto quantize during Jam
* Convert Jam session to a loop (for editing and reuse)