A jump pack is a piece of equipment used by Space Marines and Chaos Space Marines to enhance the mobility of an individual or squad, especially Assault Squads for the Space Marines and Chaos Raptors and Warp Talons for the Chaos Space Marines. It's basically a pair of jet engines that get strapped to the back of a suit of powered armor (in fact, that's exactly what the old Mk. II and Mk. V jump packs look like) to let the wearer jet around
Generally, a jump pack is used in two modes. The first is to safely descend from a great altitude, like a parachute but louder. Captain Titus of the Ultramarines (along with Leandros and Sidonus) used a jump pack in this way in the opening sequence of Warhammer 40,000: Space Marine to drop from a damaged Thunderhawk to the deck of an Ork gunship. This is represented on the tabletop by giving all jump units the Deep Strike rule.
The second mode is to move on the battlefield faster than usual infantry -- either to just cover ground faster, or to blast into close combat. Titus used a jump pack in this way towards the end of the game, jetting around the ruins of Graia and pauldron-checking Orks. On the tabletop, jump units can either move further than normal infantry (including moving over intervening units and terrain) or re-roll the dice for determining assault distance and gain some extra "Hammer of Wrath" attacks to represent them crashing into the enemy.
Only Space Marines, exceptional Sisters of Battle called Seraphim and the Orks referred to as "Stormboyz" are capable of utilizing a Jump pack due to the device's bone-shattering features that only they could survive. The force exerted by the Jump Pack while taking off is so tremendous that a normal human, without power armor to evenly distribute it, would have his spine liquefied upon take-off and the bone-shattering descent is the equivalent of a military-grade explosive going off, something that a normal human being cannot hope to survive without having his entire lower body explode in a shower of gore. In the case of the Stormboyz, they do not use jump packs so much as rokkits crudely strapped to their backs, relying mostly on their natural resilience to keep them safe (not to mention their tendency to land directly on their targets' heads).
As depicted in Space Marine, a Marine coming down with a jump pack straight onto someone's head results in a hilarious shower of gore, which is kind of explained by the Deathwatch core rulebook. On page 28 it states that a Marine in full armour weighs about 500kg, and the armour is 180kg of that. Now, imagine that kind of weight, crashing down straight onto you, accelerated downwards by a jump pack.
Units capable of an actual flight rather than quick and bone-shatteringly fast rocket jumps, such as Swooping Hawks, Scourges, or winged Tyranids have the same mechanics table-top wise, despite being much different fluff wise. You may start bitching about it.
In Warhammer 40,000 jet packs are different from jump packs. A jet pack can be used to Deep Strike into battle, like a jump pack, but it incorporates extra thrusters and control systems to keep the wearer more stable. This means that jet pack infantry do not move as quickly as their jump-pack-wearing counterparts, but are able to fire their weapons on the move as if they were standing still and move with greater agility. In-game, the former is represented by giving all jet pack units the "Relentless" Universal Special Rule, and the latter is represented by allowing jet pack units to move during the Assault phase without charging or making an assault. These two rules put together give rise to what is called the "jump-shoot-jump" tactic: a unit of jet-pack infantry jumps over cover, shoots all of their weapons (including heavy weapons) at an enemy, and then jumps behind cover in the assault phase, so that the enemy will have to shoot or charge through cover during his turn. For the most part, jet packs and jump-shoot-jump are associated with Tau Battlesuits. Eldar corsair jet packs are categorically better, however.