Spain's high-speed network is built around Madrid, and this page shows it: of eleven bookable journeys, five start there. The measured times are what make the geography workable — Madrid to Seville and Madrid to Malaga in two and a half hours each, Madrid to Barcelona in two hours thirty-five, and Seville to Cordoba in just forty-five minutes.
Two names cover almost everything. AVE, short for Alta Velocidad, is the high-speed network between the major cities; Renfe is the national operator, running both the high-speed services and the regional ones. The distinction matters mainly for frequency: the trunk routes from Madrid to Barcelona, Seville and Valencia run several times a day and sometimes hourly, while secondary and regional lines are thinner and reward checking the timetable before you fix a plan.
One practical difference from flying. Every high-speed ticket carries an assigned seat, so there is nothing to gain by arriving early to claim one — but major stations do run bag checks, so arriving early is still worth it, for an entirely different reason.