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Originally and chiefly British English, the noun bouncy castle, and variants such as bouncy air castle, designate a large inflatable structure, typically in the form of a stylised castle, house or other building, on or inside which children can jump and play.

This noun occurs, for example, in a review of the Toyota Land Cruiser, by Geraldine Herbert, published in the Sunday Independent (Dublin, County Dublin, Ireland) of Sunday 1^st June 2025 [page 36, column 2]:

Enter: the Toyota Land Cruiser, this week’s test car with seven seats, serious off-road credentials, and a boot vast enough it could double as extra storage space. The irony? No camping trip, no house move or clear-out—instead just a couple of shopping bags rattling around.
Still, it was reassuring to know that, on the off-chance I was needed for a Communion, Confirmation or emergency children’s party, I could’ve transported a grand piano, a llama and a bouncy castle without so much as folding a seat.

The earliest occurrences of bouncy castle and variants that I have found are as follows, in chronological order:
—Note: It seems that the type of structure that this noun designates was introduced during the International Ideal Home Exhibition that was held at the National Exhibition Centre, Birmingham, West Midlands, in October 1976:

1-: From an advertisement for the Birmingham International Ideal Home Exhibition that was to be held at the National Exhibition Centre, Birmingham, from Thursday 14^th to Saturday 30^th October 1976—advertisement published in the Evening Mail (Birmingham, West Midlands, England) of Thursday 7^th October 1976 [page 39, column 3]:
—However, here, bouncy and castle do not yet form a noun:

Kiddies Corner. Big, bouncy inflatable fun castle

2-: From a letter by ten-year-old Nicola Jayne Smith, of Chelmsley Wood, West Midlands, from Uncle Len’s Corner, published in the Evening Mail (Birmingham, West Midlands, England) of Thursday 11^th November 1976 [page 10, column 4]:

I had a smashing half-term. On the Wednesday we went to mom’s friend’s house and we played dancing with her two girls.
On the Thursday we went to the International Ideal Homes Exhibition at the N.E.C. It was great.
We saw a massive model railway, a model circus and a bouncy castle, which I had a go at.

3-: From an article about the Birmingham International Ideal Homes Show that was to be held at the National Exhibition Centre, Birmingham, from Thursday 4^th to Saturday 20^th August 1977—article published in the Leicester Mercury (Leicester, Leicestershire, England) of Monday 1^st August 1977 [page 11, column 1]:

A circus family appear in a show called Ricko Woodie the finale of which is Fred’s Concrete Company, an hilarious act with a sledgehammer, Fred’s wife, concrete blocks, and a bed of nails. How everybody survives is a complete mystery.
Also there will be fun and games in a play area with two bouncy castles (one for under-sevens, one for overs), a roundabout, swings and a train for rides.

4-: From the following advertisement, published in the Bury Free Press (Bury St Edmunds, Suffolk, England) of Friday 26^th May 1978 [page 7, column 3]:

SPRING FAYRE
Saturday May 27^th 2 p.m.
TOLLGATE SCHOOL FIELD
BURY ST EDMUNDS
Admission Adults 10p – Children 5p
GRAND DRAW 4.30 p.m.
Side Shows include
Rifle Range – 18 hole Crazy Golf
Bouncy Castle
ALL PROCEEDS TO AID 1^st BURY ST EDMUNDS
(MAYORS OWN) FUNDS

5-: From the Lincolnshire Echo (Lincoln, Lincolnshire, England) of Wednesday 1^st November 1978 [page 1, column 2]:

A Saxilby man was beaten and kicked on the head when he tried to rescue his eight-year-old son from an amusement park incident in Blackpool.
[…]
The incident began when Mr. Harbord’s son, Justin, was playing on a bouncy, inflated castle.

6-: From an article about the seventh Westbury-on-Trym carnival that was to be held on Saturday 19^th May 1979, published in the Evening Post (Bristol, England) of Wednesday 16^th May 1979 [page 15, column 6]:

Over 50 sideshows and stalls will be present, as well as Punch and Judy, donkey rides, a bouncy air castle, chairoplane, miniature steam train rides, a fair organ and many others.