SMS stands for Short Message Service and is also commonly referred to as a "text message". With a SMS, you can send a message of up to 160 characters to another device. Longer messages will automatically be split up into several parts. Most cell phones support this type of text messaging.
MMS stands for Multimedia Messaging Service and this type of text messaging is an evolution of the SMS. With a MMS, you can send a message including pictures, video, or audio content to another device. It is most commonly used to send a photo taken with a camera phone to another phone. Most new cell phones with multimedia capabilities support MMS.