Short answers, pulled from the story.
Thermae were the large imperial bath complexes of ancient Rome, distinct from the smaller balneae. They included bathing rooms, libraries, gymnasiums, food stalls, and spaces for socializing and poetry readings. The word comes from the Greek thermos, meaning hot.
A Roman bath was built around three principal rooms: the frigidarium (cold room), the tepidarium (warm room), and the caldarium (hot room). Larger complexes also included a palaestra for exercise, an apodyterium for undressing, a sudatorium for moist steam bathing, and sometimes a laconicum, a dry sweating-room said to have been introduced to Rome by Agrippa.
The hypocaust circulated hot air from a furnace through hollow spaces beneath the floor and inside hollow brick walls called tubuli lateraci. At Pompeii, the circular furnace beneath the caldarium measured more than two meters in diameter. A cascade of three boilers maintained the water supply at different temperatures.
Thermae functioned as community centers where Romans bathed, socialized, lobbied for political support, and invited friends to dinner. Emperors built bath complexes to win public favor, and wealthy citizens sometimes paid all admission fees on their birthdays to gain local goodwill.
The Baths of Trajan, Caracalla, and Diocletian all contained rooms identified as libraries based on wall niches sized to hold ancient scrolls. One record names a slave of the imperial household as the vilicus thermarum bybliothecae Graecae, the maintenance man of the Greek library of the baths. Scholars debate whether these libraries were open to the public or reserved for the elite.
The most complete surviving Roman baths are in Pompeii and nearby sites. In Rome, the ruins of the Baths of Caracalla, Diocletian, Titus, and Trajan remain. The English city of Bath takes its name from Roman baths built over natural springs there, and the Hammam Essalihine is still in active use. Pennsylvania Station, opened in New York City in 1910, modeled its Main Waiting Room on the frigidarium of the Baths of Diocletian.