How well does Elo rating estimate a player's strength? Edit (3 March 2014):

Proposal: strength is the real and rating is the model that estimates it.

Question: How well does Elo rating estimate a player's strength?

Bonus question: What happens if you take the player's rating curve into account?
It's not clear to me whether you are asking a philosophical or a quantitative question.

My answer to the philosophical question is that the Elo rating is predictive; that is, it is designed to make accurate predictions about the outcomes of chess games. To that extent, it measures strength by estimating the likelihood that a player will win against another player. It does this by using the player's past results, and it does the best it can given that information.

The answer to the quantitative question is that the "expected value" of a game between two players is correlated with the difference in their Elo ratings. For example if player A's Elo rating is 200 higher than that of player B, then player A would be expected to go about 76-24 over the course of a 100-game match.

Note: The question has changed since I answered it, so my answer makes less sense now. The title of the question used to be Does the chess rating represent a player's strength?, and the question was "I notice that somehow, we chess players associate the chess rating with strength. But is it a measure of strength? To what extent?" Please take this answer as a response to that question, not the one it has been changed to.