What is the difference between the airspace's classes (for non-pilots)?
Although Lnafziger's answer is correct, I'd like to elaborate on the purpose of the airspace classes.


Class A: This airspace is intended for high-speed, point to point travel.  That is why pilots flying in Class A must be instrument rated and in contact with air traffic control (ATC); aircraft above 18,000 feet are likely to travel quickly and may not have time to avoid each other visually.  This is the upper layer of the airspace system.*
Class B: This airspace is intended for large airports with lots of jet traffic.  The large number and size of aircraft require space for ATC to get them efficiently in and out of the airports.  Approach control services are provided.  As a result, class B areas are physically large.  It is possible for small aircraft flying under visual flight rules to operate here with explict ATC permission, though permission to enter may be denied during periods of high traffic.  Small piston-powered aircraft land at Class B airports infrequently, typically due to high landing fees and fuel prices.
Class C: Smaller airports with a mix of piston-powered aircraft and jets.  Most Class C airports are home to flight schools and small aircraft hangars as well as scheduled service by airlines.  While still busy, Class C airports have less traffic than Class B, so ATC doesn't need as much space to keep them organized.  Like Class B, approach control services are provided.  Pilots are only required to establish two-way communication to enter the airspace.
Class D: The smallest airspace type for tower-controlled airports, traffic primarily consists of general aviation aircraft.  Often these airports handle cargo, business jets, charter service, maintenance or assembly facilities, etc. as well as private aircraft and flight schools.  They also host firefighting operations, airshows, or anything else that needs facilities but would interfere with normal traffic if put at a larger airport.  Although Class D airports are small, some are very busy; others handle only a few flights an hour.  Approach control services are only provided if the airport is near Class B or C airspace.
Class E: This is the lowest level of "controlled airspace."  "Controlled" doesn't mean you have to talk to ATC, but that ATC services are available (within the capabilities of radar and radio equipment).  This is general-purpose airspace.  Aircraft flying under visual flight rules can fly more-or-less whatever they want (weather permitting), and instrument flight rules traffic operate under positive control by ATC.  There are regulations in this airspace, but they aren't onerous, and they're designed to accommodate the wide variety of aircraft and activities that are found here.  Some small airports have Class E airspace surrounding them at the ground.  Although Class E is a single class of airspace, there are tighter regulations above 10,000 feet, where there are no speed restrictions (other than the prohibition on supersonic flight over land), compared to at lower altitudes where speeds are limited to 250 knots.
Class F: See What is Class F airspace, and why is it not used in the US?
Class G: This is found near the ground everywhere except controlled airspace around airports, and occasionally at higher altitudes in remote areas (Alaska or the Rockies, usually), but never above 14,500 feet.  Most untowered airports have Class G airspace surrounding them.  This is "uncontrolled" airspace, so there is no ATC service available at all.  Radio and radar work better the higher you go, so at low altitudes you just can't expect good results.  Class G serves as a notice that ATC services are unavailable, typically due to a lack of radar coverage.  In remote areas, the same problems occur at higher altitudes (radar won't see you if you go behind a mountain, and sometimes you're just too far) so Class G is found there as well.  Most of the time, the only reason anybody goes into Class G airspace is because they're taking off or landing at an uncontrolled airport, or transiting certain areas that lack radar coverage.
Other: There are many kinds of special-use airspace, all of which serve as keep-out signs of varying levels from "please be considerate" up to "will shoot on sight."  In the US, these don't really interact with airspace classification.



* Technically, airspace above 60,000 feet is Class E, but nobody goes up there except scientific and spy/military flights.