A HDD are what you are mostly likely using right now, they are quite literally spinning magnetic plates that hold your data.
An SSD is based on an entirely different technology, Flash Storage. They are blazingly faster than HDD's with a near non-existent access time.
However the caveat is that SSD's are much more expensive than HDD's. Your standard 1TB HDD can achieve aound 11GB to the dollar, 2TB drives even better than that. Recently SSD have just managed to start toeing the line of $1/GB, with older/low performance ones starting to hit 80c/GB.
So what most people do is buy a small SSD (128GB) and put the OS, drivers and any important programs on there. Then get a HDD to put your big, unimportant files on, so music, movies and documents.
Heres a performance comparison between an SSD and a HDD. Its a relatively old video, so the performance has become even more pronounced. Also prices have dropped a lot, a 128GB drive is roughly $130.http://www.youtube.com/watch?v=-lR0XoHFU6Y