Points of Sailing
Running (before the wind) means sailing in roughly the direction of the wind, with the wind behind you (G in the diagram). A good boat can sail almost as fast as the wind when running (but never quite as fast as the wind, because the movement of the boat through the water causes drag).
Sailing on a beam reach means sailing more or less at right angles to the direction of the wind, by angling the sail to the wind appropriately (D and E).
Sailing on a broad reach means sailing at an angle to the wind, somewhere between a broad reach and running before the wind (F and H). A good boat can sail quite a bit faster than the wind on a broad reach. This is because the wind is still pushing the boat as long as the downwind component of the boat’s velocity is less than the wind speed. (A really good boat can sail faster than the wind even on a beam reach. It might help to think of it a bit like squeezing an orange pip out between your fingers: the pip ends up moving much faster than your fingers did!)
Sailing close hauled means sailing at an angle to wind, somewhat upwind of a broad reach (A and C). Yes, this really is possible! It’s not possible to sail directly upwind (B)*. Exactly how close to directly upwind it’s possible to sail depends on the design of the boat – and the loading, and the skill of the crew. (And a really good boat can sail faster than the wind even when sailing close hauled! Again, exactly how fast, and at what angles to the wind, depends on the design of the boat, the loading, and the skill of the crew.)
For an explanation about how a boat can sail upwind, see Sailing Upwind.
* This isn’t quite true. The kind of boat in the diagram can’t, but there is in principle a kind of ship that could sail directly upwind. I don’t think anyone’s actually made one, but they could. See Upwind Ship.