Hyperspace (also known as subspace) is an alternate dimension used by several races for faster than light travel accessible by means of a hyperdrive.
Entry into hyperspace is achieved when a vessel opens a stable hyperspace window. This allows a vessel to enter subspace and achieve faster-than-light velocities relative to real space whilst traveling at sublight speeds within its own layer of subspace. In this state, they can fly from one point to another in a relatively straight line. Hyperspace gives off high levels of radiation, but this is only a problem for the hulls of Wraith vessels, which are organic and unshielded.
Ships in hyperspace are undetectable to all but the most sensitive sensors, such as the long range sensors of Atlantis and Asgard sensors. However, most ships can detect the formation of a hyperspace exit window moments before it becomes visible to the naked eye and a ship emerges. The Asgard core installed on the Odyssey also gives off a signal which can be tracked in hyperspace when it is active. (SG1: "Small Victories", "Unending")
Asgard vessels can use neither their shields or weapons in hyperspace, and this seems to hold true for vessels of other races as well. No reason has been given for this, but it's likely related to the difference in physical laws, or simply that maintaining hyperspace velocities requires an enormous amount of power. This facet of hyperspace travel renders ships momentarily vulnerable to attack upon exiting hyperspace, assuming their aggressors know when and where the ship will exit. The subspace field which allows the vessel to enter hyperspace presumably keeps it from being torn apart, but vessels are still much more fragile in hyperspace than when combat ready, as demonstrated by Replicator-controlled vessels on several occasions. Ancient City-ships are an exception, as their shields can - and must - operate unaffected whilst in hyperspace. (SG1: "Small Victories", "New Order, Part 1", ATL: "Adrift")
While at its core all hyperdrive technology follows the same basics, hyperdrive technology of each race has a unique subspace frequency. Subspace interference broadcast at these frequencies destabilizes the hyperspace window of the specific hyperdrive technology, causing any vessels to be torn apart upon entering hyperspace; other hyperdrives would remain unaffected due to the differences in frequency. An Ancient scientist named Janus created a device to take advantage of this weakness. (ATL: "First Contact")
Dr. Rodney McKay and Dr. Radek Zelenka attempted to theorize what would happen if a craft like a fighter left a hyperspace bubble while in flight, McKay expected that the craft would be destroyed while Zelenka proposed that it would be able to maneouvre inside the hyperspace bubble. Lt. Colonel John Sheppard chose not to test out either theory when presented with an in-flight chance to sabotage a Wraith Hive ship. (ATL: "No Man's Land")