Is it teleportation, making a wormhole, or something else? Does it always exist and people just go into it, or does someone create it for themselves when they jump? Does it manipulate time?
It is a 'dimension of space-time', 'coterminous with realspace'.
For those who are unfamiliar with hyperspace, it is a dimension of space-time that can be entered only at faster-than-light speeds. While the most knowledgeable astrophysicists and astrogation experts admit that some aspects of hyperspace remain a mystery, it is understood that hyperspace is coterminous with realspace: each point in realspace is associated with a unique point in hyperspace, and adjacent points in realspace are adjacent in hyperspace. In other words, if you travel 'north' in realspace as you jump into hyperspace, you will head 'north’ into hyperspace too. Also, every object in realspace has a ‘shadow’ in hyperspace. For example, there is a star (or star- like body) in hyperspace at the same location as it occupies in realspace. Such 'shadows’ are potentially lethal obstacles for travelers in hyperspace, which is why pilots rely on astrogation computers to plot courses around the mass shadows, allowing travel from one point in realspace to another.
It is usually referred to as an 'alternate dimension'.
Hyperdrives allow starships to travel faster than the speed of light, crossing space through the alternate dimension of hyperspace. Large objects in normal space cast “mass shadows” in hyperspace, so hyperspace jumps must be precisely calculated to avoid collisions.
It’s not entirely clear, and the explanation varies based on the source you look at.
There are only a few things that are generally consistent about hyperspace across all major canon:
- It allows FTL travel in some way.
- For most races, it requires special technology to utilize it, which is lacking on some starships. Two notable examples of ships lacking a hyperdrive in the primary canon being most TIE fighters (with the notable exception to the exception being Vader’s TIE Advanced) and the Delta-7 Aethersprite and Eta-2 Actis light interceptors used by the Jedi in Episodes II and III (the big ring things with engines they dock with are external hyperdrive units).
- Some space-based life has evolved to utilize it. The sole example in current canon are the purrgil, seen in some episodes of Star Wars: Rebels.
- It is affected in some way by gravity wells. In most sources where this is explicitly mentioned, it’s implied that you can’t jump at all if you’re deep enough in a gravity well and that passing through a gravity well while in hyperspace pulls you out of hyperspace (the Imperial Interdictor-class Star Destroyers, seen in numerous recent sources, work on this principle). In some other cases it’s implied that you can jump inside a gravity well, it’s just really dangerous and it may not be possible to get beyond the gravity well.
Beyond that, it’s all speculation. The special effects in the original releases of Episodes IV and V for hyperspace travel look suspiciously similar to the VFX for warp speed in Star Trek. The special effects in at least some releases of Episode VI and most of the media since then look suspiciously similar to the time tunnels seen on occasion in Doctor Who. None of the movies ever get into an explanation of how it works, and I’m pretty sure none of the TV series do either.
The exact explanation in current canon appears to be that it’s an alternate dimension of time and space. Exactly what that means is unclear though (it could be referring to the compressed space produced by something like a warp drive, or it could be some parallel reality, or it could even just be an on-demand wormhole).
The primary explanation in the comprehensive legends canon appears to be somewhat similar and a bit more fleshed out, with the description being relatively consistent with what ‘hyperspace’ usually means in most modern sci-fi (that is, an alternate dimension with a 1:1 mapping of points to realspace, where physics are altered from realspace just enough in some way to allow much faster transits between two points than would normally be possible between the associated points in realspace).
Some older third-party extended universe sources have descriptions which are inconsistent with this, but they are mostly ignored in other media.
All of this is nice, but the reality is that Star Wars is science-fantasy, not science-fiction, and hyperspace, alongside most other ‘technology’ was really just supposed to be magic. FTL travel is a necessity for the type of story being told, so it has to exist in some form, but because the way it works is not actually relevant to the plot (well, in any of the original media it’s not, some extended-universe stuff actually does include stories where how it works is relevant) it never gets explained. This is one of numerous examples of Lucas doing a good job of adhering to the core principle of good drama: ‘Only include details that matter to the plot or are required to set the mood.’.
From the original trilogy, it is evident that "hyperspace" is just a name, and the travel does not appear to take place outside of normal space. There are are at least two definite indications of this.
First, note that as ships accelerate to hyperspace speeds, they see the stars stretch out, indicating that they have reached faster-than-light speeds while still seeing starlight in normal space.
Second, Han Solo's description of hyperspace navigation in A New Hope indicates that normal-space objects are still obstructions to craft traveling through hyperspace.
Han: Traveling through hyperspace ain't like dusting crops, boy! Without precise calculations we could fly right through a star or bounce too close to a supernova, and that'd end your trip real quick, wouldn't it?