If you have never seen this sport, probably the best way to imagine it is to think of a cross between hockey and lacrosse. It's played with a small ball - a little smaller than a baseball, but not as heavy - called a sliothar (pronounced shle-thar) and a curved wood stick, called a hurley.
There are fifteen players on each time, and the object of the game is to get as many points as possible. There are goalposts at either end of the field, and points can be scored by hitting the sliothar over the crossbar with the hurley (for one point), or by getting goal (into the lower part of the goalposts) which is equal to three points.
Unlike in hockey, the ball can be handled - but it has to be scooped up off the ground using the hurley; it can't be picked up off the ground by hand. When the player has the ball in hand, he can hit with the ball downfield, or can he can hand pass to his teammates. (He may not throw the ball.)
These clips explain in more detail how hurling is played.
Gaelic football looks like a cross between soccer and rugby, although it is in fact older than both of these sports. It is more physical than soccer, but less so than rugby. The ball used in Gaelic football is round, and a little smaller than a soccer ball, and a little lighter.
Like in hurling, there are fifteen players on each team, and the scoring system is also the same: one point for when the ball goes over the crossbar of the goalposts and three points for a goal when the ball goes in below the crossbar. The player with the ball can kick it downfield, either off the ground or from the hand, or he can hand-pass it on to his teammate. (He may not throw the ball.) To collect the ball off the ground, the player must scoop it up with his foot - he may not pick it straight up off the ground.
Check these clips out to learn in more detail how Gaelic football is played.