Well, with over 2000 known species of Cacti that are native to North, Central and South America there's a lot to choose from in determining what type of cactus you have. If you're into trading cacti be aware of the 100+ cacti that are prohibited from trade because they are endangered. You can make sure you're safe to trade by checking with CITES or The Convention on International Trade in Endangered Species (Appendix I) at this website: www.cites.org . Some major types of cacti are the Columnar, Barrel, Hedgehog, Pincushion, Cholla, and Prickly Pear.
Cactus come in many types and varieties. You will need to make a list of characteristics your cactus has and then do some research in book or on the web to determine which one yours is. You can find more information here: wc.pima.edu/~bfiero/tucsonecology/plants/cactus_?types.htm