I’ve got = I have; I possess

• I’ve got a dog.
• I’ve got a car.

I’ve got to = I must; I need to
(often pronounced as “I’ve gotta”)

• I’ve got to leave now.
• I’ve got to pay rent on the 1st of the month.

I’ve gotten = I have obtained/received
I’ve gotten = I have become

• I’ve gotten a dog.
  (I didn’t have one before, but recently I adopted or bought a dog)
• I’ve gotten a good grade on the test.
  (I received a good grade on the test)
• I’ve gotten divorced.
  (I have become divorced)
• I’ve gotten better at English.
  (I have become better at English)