"Achi" is the more common way to say "my brother". It is used in most situations, both formal and informal. For example, you would say "achi hagadol" (my older brother) or "achi hakatan" (my younger brother).
"Ach sheli" is a more emphatic way to say "my brother". It is often used when you want to emphasize that the person is your brother. For example, you might say "ach sheli, ani ohev oto meod" (my brother, I love him very much).
So, the main difference between "achi" and "ach sheli" is the level of emphasis. "Achi" is the more common way to say "my brother", while "ach sheli" is more emphatic.