In an Away Team mission, you choose which challenges to attempt, and decide on which crew member to use for a particular challenge. Each challenge has a skill associated with it (diplomacy, science, etc.) If you choose a crew member with the highest number in that particular skill, you’ll improve your chances of success; however, using a crew member also tires them, which will reduce their effectiveness in subsequent challenges.
In addition, some challenges have hidden trait bonuses. In the crew selection screen for the Away Team mission, you can learn all of the different traits which might be useful within the mission, but it will be up to you to figure out where those traits are used. If you bring crew members that have those traits, you’ll have an increased chance of success. On very hard missions, having crew members with the right traits can be the difference between victory and defeat.
Some paths in a challenge may require that you have a crew member with a particular trait; for example, you might need a Klingon or Borg character to complete a mission in certain ways. Having crew members available with those traits will be necessary to completely explore all the variations available within the mission.
After you complete your mission, you’ll receive credits, experience, and items as rewards. Complete specific parts of the mission to unlock rare rewards - uncommon or better items that help you make your crew even better!