Got .txt files in Google Drive? For Android, the thing you want is Jota Text Editor. The Drive app itself can't edit plain text files, which feels strange. But you can click and open the file in Jota+, edit away in a robust, smart editor, and when you're done, "Share" the file back to Drive. If you dig Jota+, and want to use it to directly edit and save files to Dropbox, Box.com, or SkyDrive, you can grab Jota+ Text Editor, and pay $5 to unlock its full cloud-syncing powers with a license key.
On iOS devices, the choices are basically limited to Quick Office Pro on the iPhone or Quick Office Pro HD on iPad. Both can connect to Google Docs/Drive to access files, and both can handle editing plain text files. It's one of many good reasons Google acquired Quick Office for its own app development.
"I'm on a browser and need to edit a .txt file"
The best solution for web-based Dropbox text file access is TextDropApp, a web-based editing tool that syncs to Dropbox and works in any reasonably modern browser. It's not free, but you can try it out with a full refund guarantee for 7 days. And the sooner you sign up, the less it costs--it's $8 per year at the moment.
If you're a Chrome devotee, you have other options, both of them free. Drive Notepad for Chrome creates, edits, and resizes text files inside Google Drive for your eyes and fingers. On the Dropbox side, I prefer SourceKit, which has all kinds of tools for programmers working on code files inside their Dropbox accounts, but which works just fine for editing plain test.
Not everybody needs to edit text files and keep them saved and secure up to the minute, but those of us who do appreciate smart tools for doing so. Do you do your own plain text file work? What do you use to get the job done? Tell us about your picks in the comments, or tweet at me, and we'll compile good answers for a follow-up.