HG (Hyperglide) is the world's most widespread freehub standard, created by Shimano. It has 9 splines (one narrower, so the cassette fits one way only) and takes 8- to 11-speed cassettes from Shimano, SRAM and others. Its limit is geometric: the smallest cog can't go below 11 teeth, because the freehub body is wider than a 10T cog. That's why modern 12-speed systems with a 10T cog use other freehubs (Microspline, XD).
If your bike is entry- or mid-level, it almost certainly runs HG. It's cheap, robust and universal for 8-11 speeds.
The cassette slides over the 9 splines and is held by a lockring; the narrow spline forces a single mounting position. It's the longest-lived, most compatible standard: nearly every entry- to mid-level bike uses it. Its only real limit is that it can't house a 10-tooth cog.
It takes Shimano, SRAM and third-party (SunRace, Box) cassettes from 8 to 11 speeds, and some budget 12-speed cassettes that start at 11T (like SRAM NX Eagle). It won't take 10T cassettes: neither high-end SRAM Eagle (XD) nor Shimano 12-speed (Microspline).
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Some 12-speed cassettes with an 11T cog (like SRAM NX Eagle) do mount on HG. Mid- to high-end ones with a 10T cog don't: they need Microspline or XD.
Almost: the 11-speed road one is ~1.85 mm wider. That's why an 8-10sp MTB cassette needs a spacer to fit an 11sp road freehub.
No. HG is the physical freehub; HG+ (Hyperglide+) describes the shift ramps on Shimano 12-speed cassettes.