Chess/The Endgame/King and Two Major Pieces vs. King

This is the simplest and easiest to execute of any checkmate.

The basic technique is called bicycle pedaling. This is a very easy way to drive the king to the edge of the board (but make sure there aren't any other pieces in the way!)

1.Ra4

This denies the fourth rank to Black.

1...Kg5

The Black king attempts to get close enough to capture one of White's rooks, a fruitless struggle which lasts the whole rest of the game, right up until the final checkmate. This is a very common strategy by the weak king, so bicycle pedaling works best if the enemy king is distant from both rooks.

2.Rb5+

This places the Black king in check (a beginning of a series of such checks, the last of which will be the final checkmate,) and he cannot flee to the fourth rank because of the rook on that rank. This is the beginning of the drive to the edge of the board.

2...Kf6

Black's only option is to flee to the sixth rank. He continues to approach the rooks.

3.Ra6+

Black is in check once again. He cannot flee to the fifth rank or stay on the sixth rank, courtesy of the rooks...

3...Ke7

...so he flees to the seventh rank and continues to approach the rooks.

4.Rb7+

Black is in check for the third time. This is the last time before the final checkmate. He must flee to the eighth rank, at the edge of the board...

4...Kd8

...which he does. This sets the stage for the final checkmate.

5.Ra8#

Checkmate! The Black king is trapped. He cannot stay on the eighth rank because of the rook on a8, and he cannot flee to the seventh rank because of the rook on b7. Note his close proximity to the rooks - and he started out on the opposite edge of the board!

If other pieces are in the way
A Black pawn, for instance. Say that there is a Black pawn on g6. (If there is a White pawn, the easiest strategy is to either push the pawn to promotion and checkmate the king in the middle of the board with three rooks, or simply let go of the pawn and checkmate the king with two rooks). With a black pawn, the approach to taking it relies on a doubled rooks attack, against which the Black king is powerless. Then the game could go like this:

1.Ra6

Black is given a choice: lose the pawn or lose his rear protection.

1...g5

He chooses the latter, taking the pawn along with him as a bodyguard.

2.Rb5

The first attack is lined up against the pawn.

2...Kg4

The Black king stays close in an effort to defend the pawn. Note that the pawn can't move again at this point because it is now pinned to the king. Even if it could keep running for the 1st rank, White's king is in the way, so it would only delay the inevitable; meanwhile the attempt to save the pawn would leave Black's king dangerously exposed.

3.Rg6

Now the pawn is caught in the crossfire.

3...Kf4

It makes no difference where the Black king goes; he can't move into the path of attack from the rook on the 5th rank. He tries to get out of his pawn's way, but it doesn't matter; the pawn can't move off the g-file and has nowhere to hide.

4.Rgxg5

Black is stripped of his pawn.

4...Ke4

Black settles for the best move available to him. He would like to take back with his king on g5, but as White's rooks are now doubled on the 5th rank, doing so would put his king in check. He cannot even wait on f4 and keep White's rooks tied up defending each other, because he is in zugzwang and must move somewhere. From this point, the mate proceeds smoothly as above:

5.Rb4+ Kf3

6.Ra5

Black momentarily derails White's plans of quickly bicycle-pedaling him to death. White responds by simply sliding across to the a-file and then resuming the attack.

6.... Ke3

7.Ra3+ Kd2

8.Rb2+ Kc1

Black has gotten within striking distance and is entertaining thoughts of picking up a rook, or at least delaying White's plans again for a moment. Indeed, if White gets greedy and tries to complete the checkmate with 9.Ra1+, Black's vengeance will be swift and sure with 9...Kxb2!. If White is still going for blood with 10.Ra2+, Black responds 10...Kxa2 and White has snatched defeat from the jaws of victory... another grim reminder about the dangers of pointless checks. Luckily for White, he instead continues with...

9.Raa2!

White goes back to the irresistible doubled rooks attack.

9. ... Kd1

Black's king would like to continue throwing monkey wrenches into the works by moving to b1 or at least staying put on c1, but is once again powerless and in zugzwang, and takes the only move available.

'''10.Rb1#.

And White delivers the coup de grace.

Two queens versus a king
With two queens instead of two rooks, the final checkmate is usually the same. However, there are a couple of checkmates with two queens that can be forced in the middle of the board, instead of on the edge.