Mass Effect Legendary Edition

Each character you can recruit to your squad has an accompanying Loyalty Mission to complete. Each main story mission you complete should trigger someone's loyalty mission to become available, and Kelly Chambers will alert you that someone wants to speak with you. Do so to trigger their mission to start, and I'd suggest completing them right away.
 

For the most part, you simply need to complete these missions to earn their loyalty (which gives them a bonus power and helps toward their survival in the suicide mission), but a few people have choices during the mission that can make the mission fail, permanently losing their loyalty. There are also two instances where you will complete a pair of loyalty missions and then the two crew members will get in an argument. You will need to resolve this situation using Paragon/Renegade options to ensure that both members stay loyal. If you side with one over the other, you'll lose someone's loyalty.

For a full guide on these missions, the fail conditions of the four missions I mentioned previously, and the two instances where you'll need to resolve disputes, see HERE.