Yesterday, 08:24 PM
There are two different technologies used on how cell phones work: GSM and CDMA. The major difference is that GSM uses SIM cards and CDMA does not - the technologies are incompatible with one another.
In the US these two technologies compete, and there is AT&T (GSM) vs. Verizon (CDMA), with the "discount" competition being Tmobile vs. Sprint (GSM v CDMA).
Unlocking a phone is entirely different, that's when a phone is purchased through a carrier and is locked to only their network (common when phones were subsidized and under contract). Nowadays I believe most phones are purchased at full price, I'm not sure how many come unlocked by default, or if you have to call the carrier and have it done -- I buy very few phones (1 smartphone total, haha), but unlocking shouldn't be a very big deal.
Also to note, there are very few phones that are manufactured with both technologies, but they do exist (newer iPhones and the Nexus 5, 6).
To answer the other questions -- you can buy any SIM card and they work on SIM card phones (it's all GSM). Not to confuse things, there are different sizes of SIM cards (which the carrier could immediately look up/give the proper size), its only affect is the size of the card/fitting in the phone itself. However, it sounds like this doesn't apply to your phone since it's probably CDMA (no SIM card).