When Tottenham signed Bale he was a left back. They moved him up the pitch as he developed and he ended up playing on both wings and later centrally as a withdrawn striker. He's been great in every position, but Tottenham ended up playing him centrally because they didn't have any reliable goal scorers.
I would imagine Real bought him to try to emulate the play of FC Bayern though, and he'll be starting on the right wing in home games and maybe the left wing in away games against good opposition.