The image linked to by @bukwyrm gives a great and simple way to work it out.
However, whether you call your aunt's sister's son a 'first cousin' is really a matter of personal family tradition and local custom.
A direct blood relative would always be considered an 'aunt', 'uncle', 'cousin', 'first cousin', 'second cousin', etc. That also extends to relatives who are related by marriage - by legal contracts. Aunts by marriage, brothers- and sisters-in-law, etc.
But when we are dealing with the descendents of people are who are not legally connected by marriage OR blood related, I think it is a matter of personal preference. You can call someone a 'first cousin' as a courtesy, but many (probably most) others would simply say 'the son of my aunt's sister'. The same goes for your aunt's sister. She's just your aunt's sister, unless you choose to call her something else. Your 'aunt' by marriage would be called an aunt by most people, but her siblings, who are not connected by any legal or blood connection are a matter of choice and courtesy.