Last thing first. The difference between an ssd and normal hard drive is the price; the SSD costs a lot more.
Seriously, a normal HDD stores data on rotating magnetic platters. The transfer speed is limited by the density of the data in bits per inch (not square inch, inch) and the rotational speed of the drive; the time it takes to find the data that you want depends on two moving parts.
An SSD stores your data in memory pretty much like what's in your USB thumb drive, only the controller is faster and the data interface is faster. The SSD can act as if it were a hard drive, but it stores the data in solid-state memory (hence Solid-State Drive) which gives insanely faster access times (finding your data), much faster read times, and faster write times.
SSDs are past their infancy but in the toddler stage. Different models and brands have wildly different performance, the drives have to be treated well (look for the guide by our tecmo34), and only Windows 7 can use one without crapping it up and making it slow down over time.
A lot of us use SSDs for system drives to improve boot and program-load times. Not many of us use them for data, because they are hugely expensive.
Edit: the article on Wikipedia about SSDs isn't that bad.