Solid-State Drives use something called 'flash memory' to store data, instead of a traditional magnetic disk. This is the exact same technology as in usb flash drives and SD cards, etc., so an SSD is essentially a giant flash drive.
In a traditional magnetic drive, the magnetic disks spin, and the read/write head has to move back and forth across the disks. This means that the drive consumes a large amount of power, and if the drive is dropped while it is on, the read/write head can hit the magnetic disk and damage it permanently.
In an SSD, however, there are no moving parts, so not only does the drive consume less power, but a computer with an SSD can be jostled around and dropped while it is on with very little risk to the disk.
The technology behind SSDs is still much more expensive than that for magnetic drives, which is why magnetic drives are still in common use.