It along with the Champion Belt signifies a person as the best fighter in the town, a sign of honor.
During the real Struggle competitions, it is handed over from the previous victor to this year's winner. In the virtual Twilight Town, Roxas obtains this trophy after defeating Hayner, Vivi, and Setzer. Afterwards, Roxas gives the red, green, and yellow gems to Hayner, Pence, and Olette respectively.
In the real Twilight Town, Seifer held the trophy until he saw how much better Sora was at fending off the Nobodies. At that point, he gave the trophy to Sora as a symbol of respect for the warrior. The trophy was left behind in Twilight Town when Sora and company left for The World That Never Was.
The trophy itself is an abstract blue statuette that houses four unique crystals. Each one is of a different color, and, according to Pence, there is only one of each. The digital version of the blue crystal was passed along to Sora by King Mickey, and thus, there were two blue crystals in the world, rather than just one. [Source: wikia]
You can download this papercraft model template here: Kingdom Hearts II - Struggle Trophy Free Papercraft Download