Karen and Bill have provided the traditional, accepted answers. And they're pretty much on target. Being the contrarian that I am, though, let me offer a somewhat different definition:
Fair Market Value is the most that an average buyer who is well-informed as to market conditions will pay for a home.
Let's dissect that.
"Average" buyer: One with average motivation and needs. Everyone's different, of course. But we really need to rule out people who are desperate to buy for whatever reason. Try to look at the market--the entire pool of buyers--to determine fair market value. Remember: We're talking "market" here. Example: Suppose 100 people go into a supermarket to buy cans of tuna fish. Most would be willing to spend, say, $1 per can. But now you have someone who absolutely needs tuna fish. It's for a special meal he/she is making and guest are arriving in an hour. How much will that shopper be willing to pay? $2? $3? $4? Although that shopper is willing to pay a lot more, the fair market value is still only $1.
"Well-informed." It's not enough that a buyer and seller agree on a price. A house may be worth (based on comps, and based on what an appraiser may determine) $200,000. But let's say a buyer, unfamiliar with the area is willing to pay a lot more. ($200,000 may not buy much in the Washington, D.C., area, but it might buy a really nice house in the Midwest, for instance.) So our D.C.-based buyer offers $200,000 for a home in the Midwest that's only worth $100,000. The seller happily agrees. Even though they've agreed on $200,000, the fair market value is still only $100,000.
On the flip side, investors make a lot of money (sometimes) by paying under market value. Often/usually, the seller is fully aware of what the supposed "market value" is. But for many reasons, the seller is willing to accept less in order to get a certain, quick, cash sale. Assume, again, that a house has a "fair market value" of $200,000. But it doesn't show well. The seller owns the home free and clear, and the seller isn't interested in spending money to fix the place up. So the seller--with full knowledge of value and market conditions--is willing to sell the property as is for $130,000. Those identical houses on either side of the property haven't just dropped $70,000 or more in value. (Though Karen's correct that the lower sales price will affect comps and probably will depress the supposed value if an appraisal is done.)
So, while an individual buyer and individual seller may agree on nearly any price for a property, I'd say Fair Market Value must have a firmer basis than that.
One other note: I'd disagree with your premise "that you should offer fair market value on a house you want to buy." I'd say: "Never pay more than fair market value." Quick example: A house has a Fair Market Value of $200,000. The sellers have priced it at $225,000 to give themselves a bit of negotiating room and because they want every penny they can get from the sale. Happens all the time. So, if a buyer comes in and offers $200,000, what are the sellers likely to do? Very possibly they'll counter at, let's say, $210,000. And then the buyers either accept $210,000, or they counter at, say, $208,000. But see what's happened? By offering fair market value on a house that's priced above FMV, they're very likely to overpay. Maybe a little. Maybe a lot.
Hope that helps.