"Goblet of Fire" just sounds cool; who wouldn't want to see a goblet of fire? For more on the origin of this name, check out "What's Up With the Title?" The Triwizard Cup is the goal for which Harry is competing. It symbolizes his reward for all the struggle and hardship he has endured as the youngest unintended champion in the Tournament.
What's interesting about both the Goblet of Fire and the Triwizard Cup is that both of them are smokescreens. They seem like they should be important in and of themselves – their names are capitalized, after all. But their primary significance is what they lead to: the Goblet of Fire is enchanted into choosing Harry as a Triwizard Tournament competitor, and the Triwizard Cup becomes a Portkey to bring Harry and Cedric to the Little Hangleton graveyard where Voldemort is waiting. These two objects are red herrings, things that look like clues but really aren't. You can easily forget about the Goblet of Fire by the end of Chapter 16. It's a means to an end, as is the Triwizard Cup.