Loyalty

Loyalty, featured in Mass Effect 2, is gained by completing a unique personal quest for each squad member. Upon completion of the quest, the player unlocks the character's fourth special ability as well as their alternate outfit. Abilities unlocked through loyalty can also be learned by Shepard using the Advanced Training research upgrade, though only one may be learned at any time.

Getting Your Squad to become "Loyal" in Mass Effect 2
After obtaining the first few squad members, the main quest line (with Cerberus) will lead you to Horizon, which is also where you will meet up with Ashley Williams or Kaiden Alenko (depending on who survived Virmire in the first game for a carryover character, or what gender your new character is).

You will be given the mission to gather additional squad members after Horizon. At that point, if you have exhausted the 'I want to talk to you' option for those members, each of your crew member will ask you for help with their personal issues/quest. Completing each of those side quest will make that respective members loyal to you, thus unlocking their fourth skills and an alternate costume. For some squad members, it is possible to make choices during their personal quest such that they do not become loyal at the end of the quest. During the these quests the squad member that it is for must be present and the secondary character can be anyone but will not affect the mission at all since the character wont have dialouge during the mission either.

DLC Note: Zaeed Massani's loyalty quest does not require you to have followed the main quest line to Horizon. If you are persistent and talk to him after each mission you complete, you can obtain his loyalty quest very early on.

Dispute Scene
At some point, two pairs of your squadmate will end up in an argument:
 * Legion and Tali
 * Jack and Miranda

This will happen as soon as each of the pairs Loyalty missions are finished, and you will be automatically brought to settle the dispute. You can choose whose side of the dispute you can defend, but the other will easily lose loyalty. To avoid this, you will need to have a certain amount of Paragon or Renegade points to break up the fight either peacefully or by coercion without having to choose a side and lose anyone's loyalty. If you do not have enough points in either Paragon or Renegade to resolve the issue, you can return to the character you sided against later after acquiring more points and attempt to pass another check against the stat, but the check will be much harder to pass.