There are two MLEs, and a related exception:
1) The "room" exception. Teams that utilize cap room get this exception, and lose the other two. Under the new CBA, I believe it starts at $4.3 million.
2) The Mid-level Exception. Teams that don't have cap room get this. It's available to all such teams, but a team using the MLE is "hard-capped" at $6 million above the luxury tax (known as the "apron") meaning a team can't exceed that amount at any point in the season. The MLE starts at $8.4 million.
3) The Mini-MLE / Taxpayer MLE. Teams that exceed the luxury tax by more than $6 million can use the mini-MLE, which starts at $5.2 million.
That's a really quick summary that leaves out a lot of nuance.