Wizards become the animals that suit them best but can’t choose.
J.K. Rowling stated in an interview that an Animagus will become the animal that suits them best, but they can’t actually choose what animal they become.
kelly_holland: When you turn into an Animagus, can you choose what animal you become? Or does this get "assigned" to you?
JK Rowling replies -> No, you can't choose. You become the animal that suits you best. Imagine the humiliation when you finally transform after years of study and find that you most closely resemble a warthog.
- World Book Day Chat (March 4, 2004)
This explains why Peter Pettigrew became a rat, a widely disliked creature - it suits his character but doesn’t seem like an animal most people would actually choose.
The tie between the wizard and the animal they become varies.
There are five known Animagi mentioned in the Harry Potter books - Minerva McGonagall, Sirius Black, James Potter, Peter Pettigrew, and Rita Skeeter. The reasons they each become the animal they do seems to vary. Pettigrew is perhaps the most obvious - a rat is a common word for a traitor, which Pettigrew was. Sirius Black turned into a black dog. The star Sirius is also known as the Dog Star, but it’s unclear if his name is the reason for his Animagus form.
Height or weight aren’t said to affect what animal it’d be.
Nothing is specifically mentioned about whether larger wizards become larger animals. From the five Animagi we know of, James and Sirius are both described as tall and have large Animagus forms, McGonagall is also described as tall but has a much smaller Animagus form than either Sirius or James, Peter Pettigrew is short and has a small Animagus form, and Rita Skeeter’s height isn’t described and has a very small Animagus form. This isn’t enough information to form a solid conclusion based on this alone, but it doesn’t seem Weight does affect the appearance of the Animagus form - losing or gaining weight will result in a thinner or fatter Animagus form.