Yes! Overleaf supports simultaneous collaborative editing like Google Docs.
Overleaf also borrows some ideas from version control systems, such as git: You can save "labelled versions", which are like commits, to make sure that even if others edit your document, you can always get back to and compare with important versions, like a 'first draft' or a version that was submitted for grading or publication. You can also access your projects via git, to make it easier to work offline.
The technology that powers our collaborative editing is called Operational Transformation, or OT for short. Your edits on Overleaf are sent back to the server every few seconds and saved. If two or more people edit the file at at the same time, the server is able to "rebase" each change on top of the other change, so that all of the clients end up at the same version. To notify clients of changes made by other clients, we use web sockets, which are a relatively new technology that allows servers to push updates to connected clients.
Stay tuned for future collaborative enhancements like real time cursor positions and coloring text to make it easy to see who wrote what.