A solid state amplifier uses transistors to amplify your guitar signal. Early transistor amplifiers produced a very clean, sterile sound that many guitar players didn't like. In addition, as you increase the signal to the amplifier by turning up your guitar, transistors clip the tops of the signal creating a fuzzy distortion. However, newer transistor amplifiers have a much warmer tone and are much less expensive than tube amplifiers. Don't forget - many of the distortion pedals use transistors to get the tones that players rave about.
Tube amplifiers use vacuum tubes (1930 - 50s technology) to amplify your signal. Tubes produce a richer, warmer sound and produce a very different distortion as you overdrive the amplifier. The distortion tends to flatten, rather than clip the signal peaks, which many players prefer. Tube amps are more expensive, though. There are not as many tube manufacturers and not nearly as much demand for tubes as when everyone had televisions and radios that used tubes. Tubes are expensive. A full retube of my 1965 Fender Super Reverb would cost more than many solid state amplifiers. Fortunately, tubes last a long time. Tube amps also use much higher voltages and use much more power than solid-state amps. For these reasons, tube amps require large power transformers and components that are rated for the higher voltages - this also raises the price.
You will also see hybrid amplifiers - these use tubes in the preamplifier and transistors for the power amplifier. You get a warmer sound than a pure solid-state amp, but save some bucks on the power amplifier.
A fairly recent arrival are modeling amplifiers. Modeling amps use a special computer chip called a Digital Signal Processor that emulates or "models" different amplifers and effects. Modeling amps aren't cheap, but you can get a truck-load of sounds for a lot less than a new Fender or Marshall tube amp.
To learn, keep asking. No one here had any knowledge when they started. There are a lot of great resources on the Internet, good magazines, and always, your local dealers. I know that sometimes I can save a few bucks by buying on-line, but I like to talk to someone and get to know them on a first-name basis. It's worth a few dollars to be able to trust someone's opinion.