To be brief a SSD is a Solid State Drive. It is the storage memory for your computer. It is like a HDD (Hard Disk Drive) composed of arrays of memory chips (much like what is on your computers RAM). The difference is that this memory doesn't lose what it is storing during power off modes. It is a replacement for a HDD and since it has no spinning disks or moving parts, it is faster in reading and writing information and more reliable in harsh environments. People want to replace them for speed during memory cycles. This speed increase (since the HDD is the slowest component on a system) makes the entire computer more responsive. SOme people also want to upgrade from older technology to get the benefits of the faster drives.
This is just a basic information. Read some of those posts and google to get a more thorough understanding.