[Tomisimo]

Here are a few more chairs and seats that come to mind. This is in no way exhaustive. Please feel free to point out more!

driver's seat - asiento del conductor
passenger seat - asiento del pasajero
backseat - asiento de atrás
booster seat - asiento elevado, asiento elevador, asiento de refuerzo, silla auxiliar, silla elevadora
car seat (for children) - silla de coche, silla de seguridad
stroller, pram (UK) - silla de niño, carriola (Mex)
high chair - periquera (Mex), trona
director's chair - silla de tijera
folding chair - silla plegable
office chair (on wheels) - silla de oficina
recliner - sillón reclinable (or just sillón)
rocking chair - mecedora, silla mecedora
stuffed chair - sillón
arm chair - sillón
wheelchair - silla de ruedas