When you're using public key encryption, it's important to be sure that the key you use to encrypt a message really belongs to the recipient (see key verification). PGP makes this a little easier by having a way to tell others "I believe this key belongs to this person, and if you trust me, you should believe that too." Telling the world that you trust someone's key is called "signing their key": it means anyone who uses that key can see you vouched for it. To encourage everyone to check and sign each other's keys, PGP users organize key-signing parties.
Analogy: It's like a networking party, where you introduce your friends to other friends.