Concise instructions for access with root are also found here http://www.reddit.com/r/emacs/comments/2adkvc/how_to_get_emacs_to_run_on_android_painlessly/. Note the link http://emacs.zielm.com/data/ seems broken. However, I tried to find a way to do this without root, and here are some results.
I tried following the GNURoot option offered in a couple of places (in addition to the current answer, http://www.linuxuser.co.uk/features/run-linux-on-android-part-1, http://www.emacswiki.org/emacs/EmacsOnAndroid), and since it was took moderate searching for someone with little Linux experience, this may be useful for others.
GNURoot lets you create a Linux build inside your Android. The 'wheezy' it offers by default is 'debian'. You can select a different build if you want to create it. Once you have created it, run it with fakeroot option ticked, you will then see console of your superuser.
A guide on installing Emacs on various builds exists here: http://wikemacs.org/wiki/Installing_Emacs_on_GNU/Linux
Running Emacs 23 turned out to be quite easy. You just type in
apt-get install emacs in the console, answer yes on the checks, and you are able to run emacs by typing emacs.
I couldn't get Emacs 24 running, but perhaps someone may be able to pick up from this and find a way quicker. If anyone is able to make it work, perhaps you can follow up, or update this post.
Emacs 24 seems much more tricky, as it is not well maintained on some builds (perhaps someone understands the situation better). This is available under 'emacs-snapshot' which can be found via the series of links http://emacs.naquadah.org/ -> https://launchpad.net/~cassou/+archive/ubuntu/emacs -> https://launchpad.net/~ubuntu-elisp/+archive/ubuntu/ppa, which should be the working version.
Combining the results from the second of the link sequence, you would need to type the following. You don't have to type sudo if you tick the fake root box on startup.
Note: You won't have to purge if you haven't got any version of emacs installed.
Note2: To gain the function
add-apt-repository, you'll have to install an extension beforehand, as said here https://askubuntu.com/questions/38021/how-to-add-a-ppa-on-a-server.
For my version this command worked to add 'add-apt-repository'.
$ sudo apt-get install python-software-properties
$ sudo apt-get update
$ sudo apt-get install
$ sudo apt-get purge emacs emacs-snapshot-common emacs-snapshot-bin-common emacs-snapshot emacs-snapshot-el emacs-snapshot-gtk emacs23 emacs23-bin-common emacs23-common emacs23-el emacs23-nox emacs23-lucid auctex apel emacs24 emacs24-bin-common emacs24-common emacs24-common-non-dfsg emacs24-el
To add this PPA:
$ sudo add-apt-repository ppa:cassou/emacs
Or the one he refers to:
$ sudo add-apt-repository ppa:ubuntu-elisp/ppa
$ sudo apt-get update
Then, you should be able to type for emacs-snapshot:
$ sudo apt-get install emacs-snapshot-el emacs-snapshot-gtk emacs-snapshot
*Or*, for emacs24:
$ sudo apt-get install emacs24 emacs24-el emacs24-common-non-dfsg
Unfortunately, neither of the ppa's offered in these links seem to be working at the moment, or I have missed a step in adding them. GNURoot does not seem an easy option to use. Perhaps someone is able to complete the sequence. I guess these kind of instructions should eventually be made into a community wiki.
More info could be potentially found here: http://www.emacswiki.org/emacs/EmacsSnapshotAndDebian.
Another useful option may be to instally the wheezyx option instead of wheezy to begin with, as it may offer a type of a mouse support via the touch interface.