How to draw this rook and pawn vs. rook endgame? According to endgame tablebases, this position is a draw:

[fen "3r4/6R1/1kp5/8/8/8/5K2/8 w - - 1 61"]

 1. Ke3 Kc5 2. Rg4 Kb5 3. Ke2 c5 4. Rg3 Kb4 5. Ke1 c4 6. Rg4 Kb3 7. Rg7 c3 8. Rb7+ Kc2 9. Rb4 Re8+ 10. Kf2 Kd2 11. Rd4+ Kc1 12. Rc4 c2 13. Rc7 Rb8 14. Ke2 Kb2 0-1


Nevertheless, I (as black) played on and marched the king and pawn down the board and won without much trouble.  The endgame tablebase gives 3. Ke2 as the losing move.


  Question: How does one draw this endgame as white?  (Is there a name for this position?)


I'm not finding the tablebase particularly educational.  It works, but I'm not sure why.
In your diagram/game, black is allowed to set up a Lucena position and therefore wins. However, with correct way of harassing the black king, it is possible to prevent that from happening. After 2... Kc5, you can utilize what's called a frontal defense by playing 3. Rg1. In the lines shown below, black simply cannot proceed. 

Note, that this defense may or may not work depending on which line and how advanced the opponents pawn is - see reference below.

3r4/6R1/2p5/2k5/8/4K3/8/8 w - - 0 2

1. Rg1 Kb4 2. Rb1+ Kc4 (2... Ka4 3. Ra1+ Kb3 4. Rc1 Rc8 5. Kd4 (5... Rd8+ 6. Kc5=) Kb4 6. Rb1+=) 3. Rc1+ Kb5 4. Rb1+=


If black moves his rook away from d-file in attempt to block your checks or harass your king, then you can bring your king in front of the pawn and set up a 3rd rank defense - Philidor position or a back rank defense (depending on how the game develops).

Also, this wiki page gives an excellent quick overview of R+P vs R endings.