In the US, both the bachelorette party and bridal shower are given before the wedding.
A bridal shower is a more formal type of party, where friends and relatives of the bride are invited (usually females only). It's usually held in the afternoon, and refreshments are served (but not usually beer or liquor). Silly games are played (word search puzzles, make a bridal gown out of toilet paper, things like that) and small prizes are awarded. Then, the highlight of the shower, the bride-to-be opens her presents. Quite often, the bride has already "registered" at a store, which means she has filled out a lengthy questionnaire listing the things she'd like to put in her new house after she's married. The gift registry usually includes things like dishes, pots and pans, silverware, bath towels, small appliances, etc. These are the types of gifts that people give at bridal showers. Some showers are quite large and can have anywhere from 10 to 300 people in attendance.
A bachelorette party is much less formal. A group of friends take the bride out before she gets married and has one last wild night. They might go to a bar, or they might meet at someone's home where they'll serve lots of liquor, maybe show some "naughty" movies, and perhaps hire a male stripper. If any gifts are exchanged, they're usually things like lacy nightgowns or massage oils or, um...."marital aids."