A moment is important because not only does an object have to have the forces in the x and y direction cancel out for it to be in equilibrium, but it also can't be twisted by these forces. For example, for the object at right to be in equilibrium, you already know that the magnitudes of Force F1 and Force F2 would have to be equal (see Free Body Diagrams and Equilibrium), because otherwise it would move in some vertical direction. There aren't any horizontal forces, so it's not going to move that way, but, looking at it, would you say that it's in equilibrium? Probably not, because this object is going to get twisted in a counter-clockwise direction. Force F2 is going to push the left side down and Force F1 will push the right side up. They are both going to exert a moment and this object is definitely not going to be stationary. So, we need to add another condition for equilibrium: the sum of the moments must be zero.
To be able to find the sum of all the moments, we'll first need to now how to find the magnitude of a single moment. This depends on two things: how far away it is from the point around which it will turn, and the magnitude of the force that causes it. This is why a longer lever works better. The farther away (and the harder you push), the more turning power you'll get from a lever. When summing moments together, there is a general convention that moments turning clockwise are positive and those turning counter-clockwise are negative. In the example above, both forces would produce a negative moment, and the addition of two negative numbers can never equal zero (sorry to disappoint you). In that case at least, equilibrium is hopeless (unless, of course, all forces had a magnitude of zero, but that wouldn't be any fun).
In Free Body Diagrams and Equilibrium, the examples we looked at had all the forces going through the same point. In those special cases, like at right, there can never be any moments because all the forces have a distance of zero from the single point. With the distance equal to zero, any moment will equal zero, and, when all the moments equal zero, so will the sum. So that's why we didn't talk about them before.