Reward Challenges are additional goals to achieve in each tomb of Lara Croft and the Guardian of Light and Temple of Osiris. As a reward for fulfilling a certain task the player gets objects like Artifacts, Rings, Amulets, or Weapons.
There usually are five challenges for each level. These mostly consist of collecting Red Skulls or finishing the level in a certain amount of time, but sometimes are a little more complicated like killing a specific amount of Enemies in a particular way. For more specific information on these tasks, please refer to the individual levels of the games.
Some Reward Challenges are not specific for each player: In a Multiplayer all players cooperate to fulfill the necessary task and get the same reward. (Except for players who previously already fulfilled the task.) Other challenges on the other hand might increase competition between the players.
A second kind of challenge are the Score Challenges, where the players have to gather a specific amount of points.