Garry Kasparov (known as Garry Weinstein in his childhood) was born on the 13th April 1963, and was FIDE World Chess Champion from 1985-1993 and PCA World Champion from 1993-1999. In 1985, he was the youngest-ever world chess champion at the age of 22.
Although he isn't quite my all-time favourite chess player (I am a greater admirer of Nigel Short, Alexander Morozevich and David Bronstein for instance) he was certainly the player that I followed the most closely while he was active, partly because of how successful he was, partly because of his ruthless winning mentality, and partly because of his entertaining playing style. Kasparov tended to favour unbalanced positions and played to keep the initiative, often at the cost of material.
He was a particularly strong exponent of the Sicilian Najdorf/Scheveningen formation, with the pawns on a6, d6 and e6, where White often gets attacking chances right from early in the game, but Black relies upon a structural advantage in order to hold the situation, while developing strong counterplay on the queenside, typically with ...Rc8 and ...b5-b4. He was also very good at using the Queen's Gambit (1.d4 d5 2.c4) as a means of slowly developing a strong attack- take for example his 15th-round game against Nigel Short in their 1993 world championship match:
Although Kasparov mostly adopted positionally-sophisticated openings, as is a neccessity at Super-GM level where computer-assisted preparation makes it easier to equalise against those 19th-century gambits, he occasionally tried out some 19th-century gambits- take for example his crushing win over Anand in an Evans Gambit back in 1995:
One Kasparov game that I particularly recall was his sacrificial win over Topalov at Wijk aan Zee, 1999:
At one stage he was a rook and piece down, but he had correctly calculated that his attacking chances against Black's stranded king on the queenside were more than sufficient. I particularly liked the retreat 36.Bf1!- it's surprising how often, amongst a series of checks, sacrifices and forcing moves, the key winning move is a quiet piece retreat that deflects an opposing piece away from a critical square.
Unfortunately during the latter stages of his career, some nervousness began to creep into Kasparov's play, which became evident in his 1995 match against Anand when he agreed draws in favourable positions in quite a few games. He struggled to break down Kramnik during their 2000 world championship match, and ultimately opted to retire from professional chess in 2005, at the age of just 41.
As someone who also follows snooker, I've often felt that Kasparov's chess career has had a lot of parallels with that of seven-times snooker champion Stephen Hendry- both have relied upon an attack-minded approach and ruthless winning mentality for success but have proved less strong at defensive play, and both retired in their early forties while still playing to a high level, and took up commentary roles. Kasparov's rivalry with British favourite Nigel Short also happened at around the same time as Hendry kept beating crowd favourite and entertainer Jimmy White.
For me, elite-level chess has never been quite the same since Kasparov's retirement, though there are signs that we're possibly moving towards a new "golden era" with the likes of Carlsen, Aronian, Grischuk, Nakamura and Radjabov battling it out with some long-term established players such as Anand, Kramnik and Ivanchuk, and there are strong signs that the trend is towards a unified world chess championship, following the damaging split between FIDE and the PCA.