There are plenty of tools available to help you identify typefaces (fonts), such as WhatTheFont and Identifont, as well as the experts in the forums at Typophile. You can always consult the FontShop TypeNavigator or even the FontBook. You can always break out your copy of Bringhurst for a list of great fonts along with their histories.
But let’s say you want to get better at identifying fonts on your own.
Start with the overall structure. Obviously, sans-serif, serif, and script types are different from each other (with very few unclassifiable hybrids) so you can start there. Then move on to the overall shape: tall, wide, squarish, rounded, etc. Do the letters have high contrast (contrasting thick-and-thin strokes) or are the strokes the same weight? Does it have slab serifs? Is it heavy or light?
Look for letters that are distinctive among most typefaces. Lower-case “a” is fairly distinctive in most fonts. Lower-case “o” doesn’t have much variation even from serif to sans. Some good letters to look for are a, f, e, g, k, A, T, G, K, and R.
Look for letters unique to that particular typeface. Let’s look at Hoefler & Frere-Jone’s Requiem. It has a couple of letters that are pretty distinctive. The capital “R” has a uniquely curved leg, and the tail on the capital “Q” has a pretty memorable flourish. Also, the bowl of the lower-case “p” crosses the stem ever so slightly. The bowl of the capital “P”, on the other hand, doesn’t connect with the stem at all, which is characteristic of many typefaces that originated in the Renaissance or are otherwise inspired by Renaissance types.
Now the next time you’re sitting in traffic with your designer buddies, you can play “Name That Font” with each billboard. However, it’s not recommended with your non-designer friends, as it will drive them crazy.