1. e4 d5
2. Bd3 Nc6
Not a very good opening choice for him. 2... exd5 3. Bxd5 4. Nf6 is a nice start for Black.
3. Nc3 d4
4. Nc3e2 e5
5. Nf3 Bd6
6. O-O h6
7. b3 Nf6
8. Bb2 Be6
9. c3 dxc3
10. Bxc3 O-O
11. Bb5 Bd7
12. d3 Re8
13. d4 exd4
14. Nf3xd4 Nxd4
15. Nxd4 Nxe4
16. Bb2 Bxb5
17. Nxb5 a6
18. Qg4 Nf6
19. Bxf6 Qxf6
20. Nxd6 Qxd6
21. Rf1d1 Qe6
22. Qf3 c6
23. Qc3 Ra8d8
24. Re1 Qd5
25. Rxe8+ Rxe8
26. Qb4 Qe5
27. Rd1 c5
28. Qd2 b6
29. g3 Qe2
30. Qd6 Re6
31. Qd8+ Kh7
32. Qd3+ Qxd3
33. Rxd3 f5
34. a3 Kg6
35. Kg2 Kf6
36. Rd7 g5
37. h3 Ke5
38. b4 c4
39. Rc7
http://www.eddins.net/steve/chess/ChessImager/ChessImager.php?fen=8/2R5/pp2r2p/4kpp1/1Pp5/P5PP/5PK1/8
Right up to here I think you're playing fine. I can't believe he traded off all of his pieces down a pawn and didn't even put up a fight. Not a very good strategy for White. So taking stock at the position here, Black has an advanced passed pawn, a centralized king and a rook that isn't passive. The only positive for White right now is that his rook is active. But he doesn't have much, if any, counterplay. The right move here is 39... Kd4. With one stroke you get your king defending the passed pawn and cut off the White king from participating on the queenside. The move you chose isn't losing, it just doesn't look best to me. This is winning for Black.
39... b5
40. Rc5+ Kf6
41. Kf3 Rd6
42. a4 Rb6
Remember all those positives I listed for black 3 moves ago? They're all gone now. Black has decentralized his king, made the rook passive and allowed the White king into the game. We're back into possible drawn territory here. One thing you can't be afraid of is saccing a pawn here. Take this as a potential line 40. Rc5+ Kd4! 41. Rxf5 c3 42. Rc5 Kd3 43. Rd5+ Kc2. You've allowed a little counterplay for White, but the c pawn pretty much queens by force and the White king is powerless to do anything. Remember that the value of the king in the endgame is more than a minor piece. It's approximate value is ~4 pawns. Not quite as good as a rook, but better than a bishop or knight. In this analysis position, you're effectively up over a piece even though the material is equal. Not to mention the value of a passed pawn on the 6th rank. Plus your king can gobble the queenside pawns if necessary. This is just crushing for Black.
(analysis diagram after 43...Kc2):
http://www.eddins.net/steve/chess/ChessImager/ChessImager.php?fen=8/8/p3r2p/1p1R2p1/1P6/P1p3PP/2k2PK1/8
43. axb5 axb5
44. Ke3 Re6+
45. Kd4 Re4+
46. Kc3 Re5
47. Rc6+ Kg7
Now you're the one playing without his king. 47. Re6 offering to trade rooks and keeping your king somewhat close to the action is possible here. If he trades, he loses. If not, you can think about swinging your king around to his kingside pawns. His king won't ever be able to penetrate on your queenside.
48. Kd4 Re4+
All these checks are worthless. You're just letting him march his king to your queenside and win the pawns when you should be threatening his kingside pawns. Tempos in the endgame are way more valuable than in the opening or middlegame.
49. Kc5 Re2
50. Kxb5 Rxf2
51. Rxc4 Rf3
52. Rc5 Rxg3
53. Rxf5 Rxh3
54. Kc4 Kg6
55. Rf2 Rh1
This is an interesting position... I'm not sure I can summarize this at a glance. White needs to get his rook behind the pawn before Black does and Black needs to get the kingside pawns advancing. And if the Black pawns advance far enough, he can sac his rook for the b-pawn. 2 connected passers on the 6th rank are worth more than a rook. I'd rather be Black here, but it's kind of scary.
56. b5 Rb1
57. Kc5 h5
58. Kc6 g4
I think Black is winning again, but he needs to keep pushing those pawns while he can. If they are on the 6th when he trades the rook for the pawn it looks good for him. The White king will be too far away to help.
59. b6 Kg5
I think you need to keep pushing pawns here as they are your trump card. 59.... g3 60. Rg2 h4 61. b7 Kg5 62. Kc7 Kg4 looks pretty bad for White.
(analysis diagram after 62... Kg4)
http://www.eddins.net/steve/chess/ChessImager/ChessImager.php?fen=8/1PK5/8/8/6kp/6p1/6R1/1r6
60. b7 h4
61. Rf8 g3
62. b8=Q Rxb8
63. Rxb8 Kg4
63... h3 gives you your 2 connected pawns on the 6th rank. I don't see how he can stop you. He can check you a couple of times before you get shelter in front of your pawns and if he threatens one with the rook you just push.
64. Kd5 g2
65. Ke4 Kh3
66. Rg8 Kh2
67. Kf3 h3
68. Kf2 Kh1
69. Rg3 Kh2
70. Rg5 Kh1
1/2-1/2