The class indicates transfer rate. So a Class 10 SD card should be able to transfer data at 10MBs. Since the manufacturer rates these themselves, there is no guarantee this is accurate, but it at least gives some indication of speed.
Is a Class 10 SD card really necessary? Not for every user. If you are just using it for storage to keep music on, for example, the only difference will be that saving the data to the SD card will go more quickly. However, if you are taking very high quality photos with a good camera or recording HD video to it you need to get a faster SD card to prevent your camera from downgrading your video and recording it at a lower quality.
Class 10 card aren't that much more expensive, so I always get them even though it's overkill for most of my use.