After the 1995 match and his win against Anand, Kasparov was still world champion.
But, his PCA simply stopped existing in 1996; therefore, he had to search for other ways of funding and organize his matches. In 1998, Kasparov announced that based on the ELO rating, Anand (2nd in the world), and Kramnik (3rd in the world), should play a match to determine his challenger. But Anand refused because he participated in the FIDE cycle. The match was organized between Kramnik and Shirov in 1998 in Cazorla, Spain. Why Shirov? Because he was the 4th on the rating list at that time and the logical choice.
Despite being the underdog, Shirov won with 2-0, with 7 draws, no defeat, and became the challenger! Then, Kasparov and Shirov tried to organize a match for the world title, and throughout 1998 they were in search of possible sponsors. Negotiations failed, and finally, in early 1999, Kasparov abandoned the idea of a match against Shirov. So much about the champion's right to choose his opponent. Maybe here we see some resemblance of the "old times," when the champion could choose against whom and when he would defend his title.
Then, Kasparov negotiated with Anand to play a match because Anand was 2nd on the FIDE ELO rating list, but when that failed too, he chose Kramnik (who was next on the ELO list).
This time negotiations were a success, and the match was set up for October 2000. It was named Brain games World Championship match. You can imagine that Shirov was not happy. And rightfully so.
The match in October 2000 was played in London, UK. It was set as best of 16 games, and it was played from the 8th of October till the 4th of November.
Kasparov was the heavy favorite, of course, but Kramnik stunned him with a fantastic win: 2-0 with 13 draws, no losses! It was a huge upset, and the chess world was shocked. Kasparov lost his title after 15 years of reign.
Kasparov said that he lost the title mainly because Kramnik had superior opening preparation. The thing is that Kasparov failed to win a single game! With white, he was playing 1.e4, but Kramnik put the "Berlin wall" in front of him, a variation of Spanish defense, famous for solidity. Also, after this match, that very line grew immensely in popularity, and everybody who wanted a draw or at least a solid position with black started to use it.
That same year, a huge knockout event was held by FIDE, like a world championship tournament. And in the finals, Anand managed to beat Shirov 3-0 with 1 draw (it was best of 6). At that moment, we had 2 champions – FIDE world champion Anand and Brain games world champion Kramnik.
In the following years, we saw few more cycles by FIDE and only one more match from the "other side": in 2004, Kramnik defended his title against Peter Leko. When he had to win on demand the last game to hold his title in a very tough match, he did!
At the same time, in 2004, Rustam Kasimdzanov had become the FIDE world champion.
In 2005, FIDE held a round-robin tournament of 8, in San Luis - Argentina to decide the new world champion, which was won convincingly by Topalov, ahead of Anand and Svidler.
After that, a unification title match was agreed upon and played in 2006 between Topalov and Kramnik. And this match is the subject of the following video.