Galaxy S3 Mini - my provider only offers micro and nano SIM cards
A Mini-SIM is the same thing as a standard SIM nowadays. The term was coined back in the day when SIM cards were the size of a credit card. Thus, the current-day regular SIM was known as Mini-SIM. Micro-SIM is the next step (used in newer smartphones, examples from GSMArena) and then there is the Nano-SIM (used only in iPhone 5, iPad mini and Asus PadFone Infinity, as far as I know).

See GSM SIM card evolution for graphical representation.