To view your Commendations. Click on your gamertag while in a lobby, then go to Service Tag > Commendations. Commendations are challenges, and each portion of the game has Commendations for that particular game mode.
As you progress, for example, in the "Pinpoint" Commedation in campaign. You will progressively get it to a higher level by completing more and more of that specific Commendation.
The commendations upgrade in this particular order:
Iron > Bronze > Silver > Gold > Onyx
You can check your progress on all of the Commendations online through bungie.net's Commendations Page.