SMS - 160 bytes of text each. Sends messages for "free" using the control channel. Some phones will allow you to type past the limit and send out two or more messages instead.
MMS - More like e-mail. Instead of using existing cellular infrastructure (like SMS) it will send the MMS data to your cell phone provider's MMS server which will then either forward it to another handset's MMS client or to another operator's MMS server. When the MMS is available for pickup your cellular company sends a special SMS message which your phone reads and opens with the MMS client which connects to the MMS server and pulls down the "actual" message content.
MMS can include most types of data your phone is able to open - text, images, sound, etc. It is often more cost effective to send an MMS than an SMS but that is largely due to the insane pricing structure of SMS messages.