SMS is a text message, MMS is a multi-media message, meaning a picture, sound, or video file.
SMS is (sometimes) limited to 160 characters on the receiving device, and longer messages may not be received, may be only partially received, or may be scrambled.
MMS gives you 1000 characters, plus your picture, sound, or video file, so is more flexible. Not all phones receive MMS though, and some (AT&T) do not include MMS in all of their unlimited texting packages.
Great answer by SuzyQ. Only one thing to add is that you make sure your plan includes text and picture messaging. Some plans do not include it, so you will not be surprised with a high bill.