Elo rating and fide rating I am an amatuer chess player and I want to know the difference between Elo rating and FIDE rating (for example, a 2500 Elo rating is needed for GM title).
The Elo rating system is a rating system for two-person, zero-sum games based on the assumption that performance is normally distributed. That is, a player's performance is expected to follow a normal distribution. For FIDE the mean = the player's rating and standard deviation = 2000/7 = 285.7. 

It was developed by Arpad Elo and first implemented by the USCF at a meeting in St Louis (where else?) in 1960. So, the USCF rating system is the original Elo rating system. It was later adopted by FIDE in 1970. Many other federations also implement Elo rating systems and it is also used in other sports, for instance soccer.

It has since been discovered that players' performances are not strictly normally distributed. Lower rated players have a better chance of an upset result against higher rated players than their ratings suggest and so the basic system has been tweaked by different federations (including FIDE) over the years. Varying k factors are one variable that is changed.

Elo rating systems predict relative performance, not absolute performance. Hence FIDE, USCF, Lichess, Chess24 ratings, although all elo rating systems, are not comparable. All you can say is that if player A has a higher rating in one implementation than player B then you expect that in a different implementation player A will again have a higher rating than player B although it is not certain. The likelihood will depend on the difference in ratings and the standard deviations of the systems. 

So, the bottom line is that Elo rating is a system of rating and FIDE rating is a particular implementation of Elo rating.