Solid state technology generically means, you're restricting electrons from being held in vacuum tubes or gas chambers, but has been extended to include not using magnetic storage medium.
Thus, a solid state drive uses memory cells instead of magnetic storage. The most common form of this is in electrically erasable and programmable NAND gate flash memory arrays.
While they have their own shortcomings such as limited rewrite counts due to electron tunneling, the avoid the need to have moving parts, a requirement in magnetic storage due to it's great data density and the size and cost of read/write heads.
Processor and memory performance has unfortunately progressed at a rate leaps and bounds beyond magnetic storage when it comes to input/output performance, so if you were to stick with a conventional storage device with moving parts, your processor will remain idle waiting for data from it when it otherwise could have achieved its solution quicker.
Today's commodity SSDs have data transfer rates several times higher than the fastest magnetic storage devices and can locate data in a specific location several thousand times faster.
Depending on how you use your PC, it may be extremely important, or may be almost negligible. Some operations that easily show a difference is, booting the computer, installing programs, or scanning for viruses/malware. Also, working with large amounts of data on the same drive - not copying between different drives) will also show an enormous speed up.
Additionally, if your low on memory and your virtual memory is on your SSD, the system won't slow down as drastically.
However, if you just use your computer for web surfing and checking email, you'll hardly notice a difference for the majority of your use, just slightly faster boot times and launching the web browser.