'As soon as divined service was over, the Thorpes and Allens eagerly joined each other; and after staying long enough in the pump-room to discover that the crowd was insupportable, and that there was not a genteel face to be seen..., they hastened away to the Crescent, to breathe the air of better company.' NA, Ch. 5

Since Roman times, people have travelled to Bath to drink or dip in it's warm mineral springs. In NA, Mr Allen was sent to Bath by his physician to partake of the Bath waters, and Mrs Allen is delighted to visit Bath for the society and Bath's shops.
I believe the Grand Pump Room Catherine Morland visited, was a building formally opened in 1795. It was a favourite gathering place for wealthy and socially minded people to meet, walk around and partake (i.e drink) the spring water; the waters were pumped to a fountain where an attendant filled glasses at a counter for visitors to drink the water.
Both NA1 and NA2 show a Pump Room scene; people meeting to socialize and and drink the glasses of mineral water from the counter.
The Pump Room ( you can see below in the Georgian era print) was devoid of furnishing to allow visitors to mingle and promenade in spacious elegance.
The Pump Room also kept a subscription book and new arrivals in town would enter their names. Catherine Morland found the book useful when trying to ascertain if Henry Tilney was still in twon.
Putting one's name in the Subscription Book apparently involved paying money as you were then 'entitled' to subscribe to the Assemblies and concerts in the Pump Room and to the circulating libraries and bookshops of Bath. The names of new arrivals were gathered and published in the Bath newspapers so people could read and see who fashionable visitors were in town and hope to meet them in the Pump Room.
The fashionable time to visit the Pump Room at Bath was in the morning whilst musicians played and numerous ladies and gentlemen promenaded up and down the room in social converse.
So, the Pump Room in JA's time had a socializing purpose, as much as it's purpose to dispense to visitors, the spring mineral waters of Bath.

information on the Pump Room is from JulieW; do check out he post and the picture of the Pump Room;