You cannot play GAA. The GAA is an organisation, not a sport. It governs a number of sports including Gaelic Football, Hurling, Camogie, Handball and Rounders. The rules of those sports are all different.
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For the majority of players, it is the county that they are born and live in. Players themselves don't choose to play for a county as such. Players usually play for their local club. Each county has many clubs. The best players in those clubs are chosen to play for the county team by the selectors for the county teams. So the majority of players end up playing for their own county. Even if a player moves to live in another county, they will normally still want to play for their own county. It is a matter of pride and identity, in the same way that many sports people will want to compete on the national team of their own country, if they are selected to do so. They will do their best to get to the level to be able to be chosen. GAA players are the same. They want to play for their the county that they come from.
A full set of rules, like in any sport, are complex. See the link below for more detail. Also check the videos listed.