Lost Winning Endgame - Please Help

In general the order of importance is (from highest to lowest)

1) Rook activity
2) King activity
3) Material

In particular if you can create a R+K+P trio where the pawn is on the opponent's half of the board, then that can be worth giving up a lot of material. The classical example is this game:

http://www.chessgames.com/perl/chessgame?gid=1102104

Other simple ideas can be derived from this. For example, if rook activity is important as I said, then what should white play in the position below?

(I only wanted the first move to be part of the puzzle, you can just use hints after the first move if you want)

Here's another typical example
(I only wanted the first move to be part of the puzzle, you can just use hints after the first move if you want)

Another typical decision
(I only wanted the first move to be part of the puzzle, you can just use hints after the first move if you want)

And in general, be ready for mutual attacks and threats, sort of like when players castle on opposite sides. When you're winning a king and rook endgame (or trying to draw) often both players infiltrate, both captures pawns creating passers for their side, and both rush those passers down the board and a lot will depend on efficient king and rook placement to see whose pawns queen first.

Back to our first puzzle:

In your game, though, you need to focus on not blundering. Rook endgames are tricky if you don’t know how to play them, especially giving up skewers, or tactics like how you lost. 

I don't understand starting at move 60; the position is drawn already. The correct time to deviate is move 27. b3, killing all counterplay. The pawn on a6 is already hanging, black has three pawn islands, and there is no reason to take on a6 and give up b2. 

I think 81. Ke5 is the blunder

By move 60, the game is a draw with best play. Black shouldn't experience difficulty in holding a draw. White can win with 41. d7. The white rook will next give check, and promotion will follow. 

The most important thing you can learn from this game is to look before you move! Before you played 81.Ke5, did you ask yourself will my opponent have any checks or captures? And if he does, am I ready for it? Just asking that question--before every move!--will save you hundreds of points and raise your rating hundreds of points.

You certainly outplayed your opponent in the opening an early middle game. I think 24.Qd4! was absolutely cruhing. You let your advantage slip away with 27.Rxa6 when his rook became tremendously active. 27.b3 would have retained all your advantages and you had good winning chances. 

"White can win with 41. d7."

I think you have the position wrong, or there's a typo.  41.d7 isn't legal

I was about to say you are absolutely right about 51.d7, but Mr. Stockfish, who is much smarter than I am, has a different opinion. This is actually quite interesting--at least at my level.

48....Kg5 is a serious mistake nevertheless. Try 49.d7

The last chance to win the ending was at move 50, and that because Black played badly before.

In your place, I would be extremely annoyed because of trading that "octopus knight" at d6 with Black's passive bishop at move 26. This is an elementary positional mistake.

)))