Trophies are awarded for different purposes. Some do things, others are just there! When you first earn a trophy, you will be made a small mini-cert in which you can display in your trophies proudly, or post them in your battles or journals.
Mithril Trophy: Awarded for giving a PoH to someone else.
Ether Trophy: Awarded for participating in 4 PVP Battles.
Prism Trophy: Awarded for being helpful, active, and productive in the main thread. You can nominate people for this. 
Star Orb: Awarded for winning 5 PVP Battles. +5 HP.
Love Orb: Awarded for participating in 8 PVP battles. If land on 1, +10 HP is healed.
Wave Orb: Awarded for participating in 5 boss fights. First hit from opponent does no damage. (If it misses, this still counts as a hit.)
Moon Orb: Awarded for being very MP active. May be nominated. If land on 2, It is not a miss.
If you happen to collect all of the trophies, you are able to request a growing of any pet you have without the PoH or GB. You can also bestow this upon someone else if you don't have any pets you want grown.
