Confused about t-contrasts, F-contrasts and the associated tests (t-test, ANOVA) What is the difference between a t-test and a contrast tested using a t-test?
When would an F-contrast be used instead of a regular t-contrast? Is the relationship between the two the same as between an ANOVA and a t-test?
Finally, is the answer to any of the above questions related to the fact that a "difference" contrast for a factor with N levels will have N-1 columns? (i.e., a difference contrast is only a vector (-1,1) for factors with 2 levels)
 A: Yes the question(s) could use some clarification, but here is an attempt at answering.
First you ask what the difference is between a t-test and a contrast tested using a t-test.  There are actually many different tests that can be called a t-test, this is one of the things that makes the question unclear.  Any time that you have a parameter estimate that is (approximately) normally distributed divided by a standard error (estimated standard deviation of the parameter estimate based on the same data) then the resulting test statistic is (approximately) distributed according to the t distribution and can be tested using a t-test.  This includes the 1 sample t-test, the paired t-test, the independent 2 sample t-test, test of a regression slope, etc.  Since a linear combination of normal estimates is also normal, this means that testing a single contrast falls into this category as well (the tests that I listed can be reformulated as tests on contrasts).
When comparing the t-tests and the F-tests you should understand that an F-statistic based on 1 df in the numerator is just the square of a t-statistic.  So there are many tests that could be performed using either a t-test or an F-test and you will achieve the exact same results.  The advantage of the t-test is that sometimes it is simpler and more directly interpretable and you can do one-sided tests with the t-test much more simply than the F-test.  The advantage of the F-test is that it can look at several tests/parameters/etc. simultaneously.  So just like you can compare 2 means with a t-test (or ANOVA), but need ANOVA to test more than 2 means; you can test a single contrast with a t-test (or an F-test), but need an F-test to simultaneously test more than 1 contrast and control the overall or family-wise type I error rate.
