The term gets thrown around a bit. Commonly it's the limited edition homologation GSXR750R, aka "double-R" models. They had a lot of differences from the standard issue GSXR750. Most notable visually is solo seat tail and seat-of-the-pants is an extra 8-10 hp - which came from a different head and the Mikuni "Slingshot" semi-flatslide carbs. You'll see terms like Slingshot frame, swingarm, bodywork, etc... these are referring to the double-R's unique parts. Reality, only the carbs were real Slingshot. Slingshot is the name Mikuni gave the carbs with semi-spherical ridges on the slide faces smooth out air flow.
'88-'89 standard GSXR's got "Slingshot" style carbs, too, but they were only36mm, vs. the double-R's 40mm. In '90 the standard version got 38mm "Slingshot" carbs. These GSXR's will sometimes be referred to as Slingshot bikes, too... just less powerful than the double-R version.
Here's a '89 GSXR750R:
And a standard '89 GSXR750:
And a standard GSXR750 with a cowl over the passenger seat:
(not to be confused with the solo only double-R)
The Slingshot style carbs have since been used on many other bikes, including GSXR400's, DR650's, RF900's, GSXR1100's - and the B12 as well as many non-Suzuki applications.