It has been said (and I agree) that the main difference between non-titled players and titled players is the found in the transition between different parts of the game. There are four different phases of a chess game:

1. The Opening: Piece development and King safety should be heavily emphasized in this phase; develop some plans of how to develop optimally.
2. The Middlegame: Plans of pushing advantages (or eliminating disadvantages), exchanging beneficially, inflicting weaknesses on the opponent's position, restricting opponent's plans, tactics, attacks, sacrifices, etc. dominate this phase
3. Simple Positions: Many pieces have been exchanged but it's still not quite safe enough for the King to come out and become the fighting piece he does in the Endgame.
4. The Endgame: You know the position is an endgame when your King can (and should) come into the position and start fighting.

Sometimes games go straight from the middlegame, or even the opening, straight to the endgame.Sometimes games do not make it past the middlegame or opening (early mating attacks for instance). But these transitions are where many mistakes are happening, because the player will need to adapt to an entirely new set of principles. In the opening and most middlegames, it's important to make "fast" moves; moves that develop along the plan the player has deemed beneficial to their position. If you are not "fast" in these phases your opponent may checkmate or stop your plans. 

In simple position and the endgame it's more important to play the most beneficial moves to your position, no matter how long your plan takes. Your opponent will most likely not be able to launch a quick strike against your King in these phases; so there is less need to play "fast". 

Many students complain that a huge majority don't make it to the endgame so they feel less inclined to study this part of the game. It is true that many games, especially at the club level (1200-1800 elo), do not go to an endgame. This does not mean that endgame knowledge is not necessary though. Many of these same students show me their games and in these games there are plenty of opportunities for the student to trade down into a clearly winning (or clearly drawing if they were worse) ending. So it's true that that game didn't end in the endgame, but that's not how it should have ended. The ending is the phase where advantages are supposed to be cashed in, but this is impossible if the student doesn't know the technical aspects of the endgame. Why are we studying only Rook endgames? Here's some stats: 

All Rook Endings                 61.90%  1 or more Rooks present 

All Bishop Endings               35.44%   1 or more Bishops present

All Heavy Piece Endings       29.74%  Queen/s and/or Rook/s, no MPs* with/without pawn/s                                                                

All Queen Endings                 9.29%   1 or 2 Queens present 

All piece vs pawn/s endings    5.83%  (inc. all one piece vs pawn/s endings)

Unspecifiied MP Endings         4.69%  (R vs B/N, Q+MP vs Q and Q vs 2/3 MPs)

Bishop Only Endings            7.10%   inc B vs Pawn/s     

King    Only Endings            5.83%   inc K+P vs K         

Knight Only Endings            4.67%   inc N vs Pawn/s     

Queen Only Endings            3.69%   inc Q vs Q + Q vs P endings 

These stats are from: http://www.chess.com/blog/NimzoRoy/endgame-percentages

As you can see, a great majority are Rook Endgames, so studying this type of endgame will help you the most. Also Rook Endgames have some very challenging theoretical things you must know (you most likely will have no chance to find it over the board) in order to survive or push through an advantage. 

I'll get off my soapbox, but please believe me that adding technical knowledge of endgames will greatly improve your play. Below are some puzzles and games to teach some technical aspects of the Rook Endgames and also some puzzles that show that studying Rook Endgames can be fun!

Important Rook Endgame Principle: 

It's Better to Have an Active Rook than to Passively Guard a Pawn

From the beginning position of the last game, find the active defense that would have lead to a pretty easy draw for Black.

Just in case you don't believe me, here's an example I played out on: http://www.shredderchess.com/online-chess/online-databases/endgame-database.html  

At that above website play out this position, it's interesting how little changes to the position change everything!

Let's see if you can draw against one of the greatest endgame players ever, just as Yusupov did

Rook vs. Pawns:

The endgame I have the most trouble with

The Lasker Maneuver

Rook Endgames for Fun

Q: White missed an amazing win in the above game, what was it?

Additional Links pertaining to Rook Endgames:

1. For the Lucena Position, Philidor (Endgame) Defense, and Vancura Position go to: http://www.chess.com/blog/CraiggoryC/rook-endings-that-every-tournament-player-should-know
2. A great article by WGM Natalia_Pogonina: http://www.chess.com/article/view/studying-rook-endgames

    

3. A great endgame reference, a 6 piece tablebase: http://www.shredderchess.com/online-chess/online-databases/endgame-database.html