The Hedgehog System was originally in reference to the Hedgehog English in the English Opening (1.c4). But the use of the term “hedgehog” in chess openings seems to have broadened to refer to almost any opening where Black hunkers down with a passive position, but puts some “spikes out”. So if you want to say “I’m going to play the Hedgehog”, it’s not going to be entirely clear to the other player what opening you’re meaning.
The original Hedgehog English has Black playing “e6” and “b6”. Other types of system given the nickname “hedgehog” include those with “e6” and “d6”, where Black might play Be7 and Bd7. Another version would be “e6” and “d6”, but with Ne7 and Nd7. Or perhaps even “e6”, “d6”, “g6”, and “b6”, with Bg7, Bb7, Ne7 and Nd7.
Obviously the same ideas could apply to White opening setups. That would be a “reverse hedgehog”.
The various hedgehog systems are not particularly good, but are playable. They are not a common choice for master players. The problem with them is that Black makes no real attempt to claim any of the center, and so White ends up easily with a space advantage. White can then play for a slow “squish” in a “boa constrictor” style, or can venture a more ambitious attacking plan.
The upside of the hedgehog is that Black has a solid setup with no real weaknesses, so there’s not usually a quick crushing attack for White. Also, the spikes have real sharpness, so tactics can spring from the Black position if White tries to go for too much, too soon, and over-reaches.