I just joined and I am having a little trouble understanding the interface. Six projects are under the 'Projects' tab, which is the amount I signed up for. However, there are only 3 tasks under the 'Tasks' tab. What is particularly strange is that one of the projects that currently has no tasks did give me a task earlier. I have messed around with a bunch of the options, and then pressed the 'Update' button, but it did not work. I do not think that this is a bug with BOINC Manager; I am pretty certain that there must be some obvious thing that I am missing. Do you have any advice?

A couple of reasons to start the ball rolling.
First, not every project has a continuous stream of work available, thus you may have been unfortunate in your timing.
Second, BOINC may think the work you have received may have filled your cache, so no more work will arrive until you have returned some and a degree of calibration has taken place.
Third, have you got your options right on each project, some projects are very "fussy" about having the right processor and processing options set before they will send out work.
Without knowing which projects you have joined it is hard to know which of these situations apply.

In addition to what robsmith said, you may have Manager showing only active tasks, that is, tasks that are being processed right now. Tasks tab has button Show all tasks in that case.

You should also consider the number of cores your CPU has : by default boinc is setup to use 100% of CPU capacity, e.g. on my HT i7 boinc can run 8 CPU tasks at the same time, and it does.

3 active tasks at the time is really not a "default" possibility (it would be 1, or 2, or 4, or 8... depending on your core number and being hyper-threaded or not) unless boinc cannot get all the tasks it's asking for to the projects for some reason like "no task available on the projects" (this can happen often depending on the project) or "some limitation prevent your boinc from getting more tasks" (your memory / disk settings could be too low, there are some parameters about that in boinc too).

The cache robsmith is mentioning is the workload estimated duration setup in boinc parameters (number of days of calculation), by default it's not very big, you can increase it up to 10 days.