Fancy mice are the most common type of pet mice. They come in the smaller 'pet' variety most commonly found in pet shops, and also the larger 'show' type that are bred for showing as the name suggests.
Multimmamate mice are the next common - these are also known as natel rats, or african soft-furs. They are larger than pet mice and sometimes have a reputation for being more aggressive although this is usually down to poor upbringing and lack of handling whilst young. They are sometimes referred to as the link between rats and mice.
More exotic tyes of mice include African striped grass mice (Zebra Mice), African Pygmy Mice and Harvest mice. The availability of these may depend on the laws in your country, and they require more specialised care than fancy mice. They are not often handleable so are pets more to observe than interact with.
You can, but it will not be as good a pet as a domesticated mouse.