Hogwarts has a trophy room where all the old awards, trophies, statues, cups, plates, shields, and medals are kept in crystal display cases. There is also a list of all the head boys and girls.

It's adjacent to a long gallery full of suits of armor. Peeves likes to bounce around the trophy room often.

Medal for Magical Merit
Given to Tom Riddle 50 years ago. Since Tom was one of the brightest wizards ever to attend Hogwarts, this award may be one for achievement in academic studies.

Gold shield awarded to Tom Riddle for exposing Hagrid as the one who opened the Chamber of Secrets and released the monster.

Harry and Ron received Special Awards for Service to the School after they rediscovered the Chamber and defeated the basilisk (and the memory-Tom Riddle as well)

House Cup
Won each year by the Hogwarts House that has earned the most House Points.

Quidditch Cup
Won each year by the top House Quidditch team

Triwizard Cup
Awarded to the winner of the Triwizard Tournament and held as an honor by the victorious school until the next tournament.

Gryffindor Quidditch shield
A shield is shown with several names on it. At the top it says "Gryffindor" and prominently displayed in the center is a small shield with James Potter's name on it, listing him as playing Seeker in 1972. The other names are R. J. H. King and M. G. McGonagall. Since we know that James was born c.1960, this would suggest that he won the award early in his years at Hogwarts.

Order of Merlin
The Order of Merlin is a great honor, bestowed on wizards who have demonstrated their devotion to the principles of living harmoniously with Muggles. The Order was created by Merlin, the famous wizard of the Middle Ages, to pass laws against using magic against Muggles. It is likely that the true meaning of the Order has been somewhat lost over the years, since Sirius Black's grandfather had been given an Order of Merlin, First Class. The medal was found on the drawing room shelves during the cleaning of that room. Sirius said he'd gotten it for donating a lot of gold (OP6). Others who have received the Order include:

Albus Dumbledore - Order of Merlin, First Class

Adjacent to the Trophy room, is the Armor room... Obviously filled with, well... Suits of armor.