The Google-related solution was available here at http://docs.latexlab.org/ but has been taken offline.

Note, however, that the online compiler was never affiliated with Google. It simply used Google UI toolkit and Google Apps hosting platform. (from Dima)

Overleaf

Features:

• registeration required


• collaboration possible, by means of sharing the URL


• documents can be saved, actually everything is saved automatically


• instant compilation on-the-fly


• possibility to upload files (PDF, PNG, STY, TeX, Bib, etc.) up to 10 MiB


• syntax highlighting in the included editor

Screenshot:

ScribTeX is another good choice. You can also checkout the Common LaTeX Service Interface (CLSI) which is one of the underlying technologies of both ScribTeX and LaTeX Lab.

The CLSI provides the underlying infrastructure for exposing a LaTeX compiler to requests from the internet and can be used to build your own custom web-based solution.

Update

ScribTeX uses git internally to store revisions to documents and the site's author is planning to open the service up to allow users to push and pull repositories as soon as he gets the logistics worked out.

Looks like there may finally be a GitHub-like service for LaTeX documents!

The ConTeXtGarden offers a simple ConTeXt online compiler. It uses an up-to-date ConTeXt MkIV version (LuaTeX).

Here is a screenshot (of an older version which still offered the choice to compile with MkII):

Verbosus seems to be the most convenient and appealing to me.

It worth noting that Verbosus has a good app for smartphone and tablet (with a dark theme, which someone may prefer) and, last but not least, use an updated version of TeX Live, whereas Overleaf and ShareLaTeX do not.

On the other hand, it has not the instant compilation on-the-fly, which Overleaf has.

I’ve used some of the online compilers, and they’re fine, but occasionally I want to make use of my own TeX installation on my work computer because it has custom packages and settings installed not available with the online compiler. But what to do if I’m not sitting in front of my work computer?

My solution was to set up a SSH server on my work computer, and simply use an SSH client from wherever I am to log in, fire up vim or emacs or other text only text editor to do edits if need be, and/or simply run latex/pdflatex from the commandline through the shell.

Apparently, you can do this with an iPad or similar. (I don't have one.)

A recent addition to online compilers by Troy Henderson, presented at TUGboat 33:1, 2012 is the LaTeX previewer:

One more solution, not mentioned above: papeeria.com
It's currently in active development.
Free plan includes one private project and unlimited public ones.

SimpleLaTeX: A simple online editor for very minimal documents; manages and previews LaTeX notes:

SimpleLaTeX is an online tool where you can preview and share short notes in LaTeX. It may be useful if you are writing a complicated equation, table, or TikZ image that requires iterative trial-and-error. Current features include:

• A trimmed image is generated and displayed in scalable SVG;




• Images in PDF and PNG formats are also available for download;




• You can save your note being sketched to your browser's cache, which will be loaded next time you visit;




• You can publish your compilable note and share it with others.

The site requires HTML5 support so it may not work in older browsers.

SimpleLaTeX is not designed to handle full LaTeX documents such as Overleaf does. The goal is to be light-weight and in spirit more like jsfiddle.

Amusingly, many of the online previewers given in other answers are vulnerable to maliciously crafted input. I can only think of one fairly-far fetched way this could be a problem for users, but the people running the preview services should think long and hard about what it is they're doing.

Here's another one I found:

http://tools.jcisio.com/tex/eq.html

I've blogged about using CLSI (in particular ScribTeX) to compile LaTeX remotely. I wrote a little client in F#, but it's pretty easy to code a CLSI client in any language.

I also show conceptually in that article how to integrate this with a source control repository and a build server.

Another solution not mentioned here is Authorea which lets you collaboratively write LaTeX (and Markdown), and render it to HTML or PDF (most journal styles supported). Also- it is built on Git for version control.

I think that my minimalistic service, latex-online, might come handy to someone.

Unique feature:

• Create a live link which, when pressed, fetches content from url/git repo and returns freshly compiled pdf. Github example
  

This helps a lot if you store TeX files in a public repository and want a "See Latest" link in the README.md (example: my diploma).

Other than that, main features are:

• Compile given URL/gitrepo/text.


• Compile local files (with the help of command-line utility
  


• Open Source!


• Easy docker deployment

I find this useful - but once again, beware: I'm the developer of the thing and might be biased.

Additionally, Verbosus not only allows using LaTeX in the browser. The developers also provide an Android app called VerbTeX and an iPad/iPhone app called iVerbTeX. I guess this is definitely a nice addition to browser-based LaTeX editing.

The following is the only LaTeX online editor that I found both to be 100% free and to support collaborative editing. It was also designed for sharing bibliography, but I have not looked into that feature.

www.publications.li/

All you have to do is register and you are ready to start a document. To share the document with an "unconstrained" number of collaborators, just send them the document's URL. I think it is worth the try.

Auto-Latex Equations add-on for Google Docs

For just equations, the Auto-Latex Equations add-on for Google Docs is free and works brilliantly. Additionally, it replaces all your equations with images of the high-quality equation, making it mobile-viewable and fully compatible with Google Docs image tools.

All you have to do is type an equation within delimiters, like $$55 + sqrt5$$ and it can be rendered in super high quality at whatever time you like by rendering all the equations in your document. If you mess up, you can always undo one or all the equations. It supports collaborative editing, but not the full LaTeX document syntax.

You can get it for free at the Google Docs add-ons store.

Texpad developers here. I should add that remote/cloud typesetting has become increasingly unnecessary as on iOS we have our universal (iPad/iPhone/iPod Touch) app called Texpad that has the most complete local LaTeX typesetter with advanced fonts, Beamer and tikz.

Hope it helps to the OP.

For the MetaPost users, besides Troy Henderson's LaTeX previewer already presented above there is also its MetaPost previewer: