The pairwise comparisons test used by default with multcompare is Tukey-Kramer. Using the ctype option, you can change the test to Bonferroni, Scheffe, Dunn-Sidak, or Fisher's LSD. Generally, the different tests yield the same or similar outcomes.
The discussion on t-tests in "More About" has nothing to do with how the tests are done in multcompare. This discussion, which I just read, is a nicely written explanation on why a customized test, such as Tukey-Kramer, is needed in the first place. The key point is this: When there are many group means, there are also many pairs to compare. If you applied an ordinary t-test in this situation, the alpha value would apply to each comparison, so the chance of incorrectly finding a significant difference would increase with the number of comparisons. All the pairwise comparisons tests accommodate the latter issue (incorrectly finding a significant difference) by applying a more conservative criterion than would exist if a simple t-test were used for each pairwise comparison.