Are saints and martyrs the same thing?

Nope.

Saints have gone through the process of canonization. This is a lengthy process, notably used by the Roman Catholic Church and various offshoots, that seeks to prove the person in question is acting as an intercessor in some fashion (e.g. answering prayers, performing after-death miracles, holy visitations, etc). A martyr is someone who has died for her/his religion and is recognized by that religion as having made that supreme sacrifice. Almost anyone can become a martyr since all it requires is getting killed for the religions. Becoming a saint can take decades or centuries.