Install AIDE from Google Play on your Android device (all Android versions from 2.2 are supported) and start it. AIDE comes with a built-in Git client, so there is no need to install anything else.
To clone an existing Git repository open AIDE's file browser, go to the folder where you want to clone the repo to, and select "Clone Git Repository here ...". You will be asked for the repository URL and the new repository folder name.
Depending on the authentication method of the remote repository you may need to setup SSH keys. AIDE supports open SSH key files. Put them in a directory on your SD card and open AIDE's settings, where you can configure AIDE to lookup keys inside this directory.
When you create a new app project AIDE provides the option to create a new Git repository for it. This is usefull even if you don't push it to a remote repository immedeately, since it allows you to better track your changes. You can push a newly create repository to a remote repository at any time. AIDE will ask you for the remote URL when you select "Git > Push..." for the first time.
To commit changes open AIDE's integrated file browser and select "Git > Commit...". AIDE will show a list of changed files as well as a diff showing the changed lines inside each file. Select the files you want to commit and tap "Commit". You will be asked for a commit message. Enter something meanigfull here. You can also discard changes to the selected files by tapping "Discard". Git is a distributed version control system, so committed changes will not be pushed immedeately to a remote repository. You have to use "Git > Push..." to push your commits. Note that comitting changes is only possible if you have the AIDE Premium key installed.
AIDE supports working with Git branches. You can checkout an existing branch, create a new branch, delete a branch, and merge a branch into another branch. Newly created branches can be pushed to a remote repository as well. Branches are a convenient way to do experimental changes to an app with AIDE without having to commit the changes to the main code base. Get more info about Git branches here.