How can I find out how often my publications have been cited? Is there a special database somewhere where you can see how often other scholars have cited you or your publications in their scholarly works? 

My professor is always saying, "No one has cited me on this..." 

Is that just his general assessment or does he know this from a verified source?
Google scholar works pretty well, though with all of these things there are edges cases that cause problems (non unique names, random poster presentations showing up counting as citations, etc.)

I find that ResearchGate tends to be pretty good. People can comment, annotate, and more generally fix up the metadata issues on this site in a social media style, and so it tends to be pretty accurate. However, since it's still quite new most papers are missing from its databases (though it still tends to count citations correctly?).