Escaping from a grab is Athletics versus Fortitude or Acrobatics versus Reflex. In those cases, the creature's Fortitude or Reflex defense is the DC for the Athletics or Acrobatics check.
An opposed check is different, and rather uncommon in 4e. It tends to come up when two characters are fighting over the same thing. For arm wrestling, you might make opposed Strength checks (each creature makes a Strength check; highest check wins). You could do something similar if they're competing to solve a puzzle with Intelligence checks, or seeing who's better at darts with Dexterity checks. You could have an eating or drinking contest be done with opposed Endurance checks.
A common use of opposed checks comes when one creature tries to hide from another. The hiding creature makes a Stealth check with a DC equal to the passive Perception of the one it's hiding from. If the creature becomes hidden, its opponent can make an active Perception check against the Stealth check that the hider made to hide. It's an opposed check with a time gap (I make my Stealth check now; if it beats your passive Perception you make an opposed Perception check later against my earlier Stealth result).