During the events on Virmire you’ll have to either convince Wrex to stand down, or put him down.
Choice: Kill or Calm Wrex
After you link up with Captain Kirrahe on Virmire, you’ll find out that Saren is breeding Krogan at his nearby base, having figured out how to cure the genophage. Not surprisingly, Wrex isn’t happy about the Salarian’s intention to destroy the base, nor your inevitable complicity in the act.
There are a few ways this encounter can play out, but for our purposes only two matter: Wrex dies, or he remains loyal. In the former case you can either electively put the krogan down on a whim, otherwise Ashely will intervene if directed, or if the conversation carries on too long.
Keeping Wrex alive involves earning his trust or succeeding at a Charm/Intimidate check. The Charm/Intimidate checks are fairly high (8 ranks each) and unfortunately earning Wrex’s trust is something that has to be done before you land on Virmire. You can achieve this by completing Wrex’s quest Wrex: Family Armor. If you completed this assignment ahead of time, you’ll be able to calm Wrex down just by picking the top option every time - no Charm or Intimidate needed. Specially, be sure to pick “I wouldn’t do this otherwise.” and “We are.” when you get the chance to convince Wrext to stand down.
If you manage to resolve matters with Wrex by using Charm or Intimidate, you’ll earn the Charismatic achievement.
ME1: Use Charm or Intimidate to resolve an impossible situation in the game.
If Wrex dies on Virmire, he obviously won’t be a squad member anymore, depriving you of his services for the rest of the first game.
If Wrex is alive in Mass Effect 2 he’ll be in charge of clan Urdnot, otherwise another krogan - Urdnot Wreav - will be in charge. This affects a bit of dialog but is relatively inconsequential as far out outcomes go.
Mass Effect 3 is somewhat more consequential. Both Wrex and Wreav will become War Assets, although the former is stronger than the latter. In addition, Wrex will also give you the Krogan Mercenaries War Asset should you genuinely cure the genophage, but if you were disingenuous about the “cure”, Wrex will ultimately end up attacking Shepard, and with Wrex’s demise, so to will die any krogan support. Wreav will not give you Krogan Mercenaries as a War Asset, but he can also be tricked by a fake cure.