It refers to controlled airspaces that require a clearance to enter, whether you're passing through or landing. In the US, you've got Class B, which you'll see around most big international airports.   The aispace is shaped like an upside-down wedding cake; from the primary airport out to about five miles it goes from a specified altitude all the way to the ground, then it lifts up to usually 500 feet, then 1500, etc. so sometimes you might be taking off from some little dirt strip underneath a controlled airspace and need a clearance to climb up into it if you want to. Once you're cleared, you are under ATC control. You must go where they tell you, at the altitude specified. In Class B and C airspaces you must also have a working transponder, so they can get you on radar properly. If you're flying some vintage model that doesn't have a transponder and you are told to "squawk" such-and-such frequency, you don't belong there.
Class C and  D airspaces (found around smaller controlled airports)  don't require a clearance, but you do have to establish communications, and they can tell you to get out or stay out if they need to. Class C airports are always radar-serviced airports, while Class D may not be. They're usually busier than Class D, so although you don't technically need a clearance, you're more likely to be denied access unless you're landing (this has been my experience in real life). They're usually smaller and lower than Class B, so going around or over them is an easier option. And if you're given heading and altitude instructions, obey them. The whole purpose of ATC is to keep airplanes from hitting each other. Period.
  Here's how it works when you enter a Class B airspace:
If you're VFR and not using Flight Following, you must contact ATC (usually the Approach frequency but sometimes the tower) and request clearance to enter or transition the airspace. You may not enter until you hear "clear to enter".
  If you're VFR with Flight Following, the controller watching you will either give you the clearance or hand you off to another controller who will (or won't).

  In FS9, I've found that when I'm IFR (or VFR with Flight Following), I just get handed off... if I'm on an IFR flight plan, I'll be handed off eventually to the tower if I'm landing. VFR to land: you need to contact the tower on your own, after you get cleared to enter the airspace, because ATC isn't tracking your plan, and won't know if you're going to land there.
A warning: in FS9, I've noticed that when I want to enter a Class B airspace, sometimes the sim only gives me the option to request clearance "into the class D airspace"; if you see that, don't worry, it's just a glitch.

If you're not sure how to tell what airspace is what in FS9 and how to tell if you're in one or not, go to the Map page and mouse over to the airspace boundaries shown. A box will open that you can click on to get the info on that airspace. Class D airspaces are not really shown (usually Class D is single-tier and extends only 5 miles around the airport), but it can be assumed that any blue airport on the map has a tower and is Class D.

As for MOAs, Warning Areas, and Restricted Areas, I stay the hell out of them in the sim and in real life, so I don't know the procedures for those. Again, you can ID those and find out the altitudes using the Map page.

One more thing: if you're using GPS, you will see the "MSG" text flashing whenever you're in or near a controlled airspace, and you can also get airspace info from the GPS. But it's easier to use the Map to figure it all out when you plan your flight...