Simplest way to jointly write a manuscript?

If your collaborator is not technically inclined, then there are a few possibilities:

• Google Docs
  It is very simple, robust and real-time (no need of thinking 'which version is the most recent'). I created a lot of documents with it (though only 1 scientific article). A good idea is to use different colours so that it is clear who wrote what. One drawback is that it may be not as simple for equations and than you will need to copy things from there to LateX manually.
• Annotating PDF files with Adobe Reader or sth equivalent
  Also simple and robust. However, as it is not real-time you need to keep track of which version is current and it is not possible to make changes in the same time. Here you are the only person responsible for changes in LaTeX file.
• Annotating or changing with OpenOffice, MS Word or another office package...
  Another variant which allows you collaborator to make direct changes, at cost of some robustness.

As a practical remark for the two last: always keep track of the version, e.g. draft_20120424_prof.pdf, where you add both date of the last edit (not final5!) and name of the last editor. When exchanging things by e-mail it is easy to loose track which version is the most recent and in fact lost changes (often not knowing about that.)

Other possibilities, requiring some technical knowledge:

• writeLaTeX or sth similar (ShareLaTeX, ScribTeX, LaTeX Lab - the one using GoogleDocs, ...)
  Tools for editing documents collaboratively (in real-time) and compile it within one's browser (so no need to install anything). Though some knowledge of LaTeX is required (at least to know how not to spoil a file).
  ScribTeX(the only one I used) gives version control and you can see which lines were changed (as in version control system), so not need of guessing what might have been changed.
  EDIT: ScribTeX is being replaced by ShareLaTeX.
  writeLaTeX has live preview and may be the most suitable for beginners.
• Git or another version-control system, e.g. at GitHub
  However, it may require technical skills beyond edition of LaTeX. While it is not hard to use it, I never did I am using it for writing papers and I would never try to use it with someone with low computer literacy.

When it comes to Dropbox - it is a good stuff for sharing files for a project but NOT for making changes in files. First, you never know when the changes happen. Second, you don't know where the changes happen.

There is also an article on it: Research tips - Online collaborative writing.

See also a few questions:

• Simultaneous collaborative editing of a LaTeX file - tex.SE
• Tools for collaborative paper-writing - MathOverflow
• Compiling documents online - tex.SE

If you're working with Word documents, Word 2007 included a very nifty merge changes feature. This online help document from microsoft.com details how it works, but doesn't really do the feature justice. See the following screenshot for an idea:

Briefly, you can click on "compare" (in ribbon, third from right), and it lets you choose two source documents. From there, it opens the very nice interface shown above, with the two source docs on the right, the merged doc in the middle, and the changes on the left (or bottom, you can hide it if you don't wan t it). You use it like any other "track changes"; just go through, change by change, using the "previous", "next", "accept", and "reject" buttons om the Changes section of the ribbon. When you're done just save the combined document, and you've got your merged doc.

I've used this a number of times, and it's a very useful feature. By far the best aspect of this is that it can be used with the technically incompetent. Just tell them, "take the document, make changes, I'll deal with it"; I've used this with my technologically challenged boss a number of times, and it's been a lifesaver. Given your situation, this may be your best bet.

It also depends on what should be the final format of your document. For instance, I write all my articles in Latex, and when collaborating with other researchers, I use BitBucket: I create a Latex repository, and give access rights to the other authors. One good aspect of BitBucket is that they have a special plan for academics, where you can get an unlimited number of collaborators for free. In order to avoid conflicts as much as possible, we usually decompose the main document in one file for each section, and we try to avoid working on the same section at the same time, but even if it happens, then it's usually possible to easily merge the different edits. One nice feature is the possibility to automatically receive notifications when one author commit her changes. Also, since anybody can create a BitBucket account, there isn't the problem of creating an account to an external collaborator on an internal server.

Of course, BitBucket is not the only possibility, and one could also use Git, or even Dropbox, although for the latter, I don't really like the lack of native versioning system.

On the other hand, if I don't work directly on a paper, but for instance on a draft for a project proposal, then I use Google Docs, because it's quite simple to do some collaborative editing.