From Halopedia, the Halo wiki
The Warthog Jump was first recorded by Randall Glass. It is commonly done in The Silent Cartographer, where you launch a Warthog into the air using grenades and/or rockets. In Halo 2, the Sputnik Skull can be used to make Warthogs fly farther than they would normally. In Halo 3, through use of the Forge the player can put a Warthog atop a pile of fusion coils; though this normally destroys the warthog if vehicles are toggled to be destructible, the wreck may fly a good distance.
Randall Glass had a way to describe the Warthog Jump wholly:
Note: Only dropped grenades will have a chain reaction. The ones you find near the flipped Warthog won't go off. Also, multi-player grenades won't work either.
Warthog jumping in Halo 2
On the beach on Zanzibar there is a ramp-like piece of debris (the ideal game type to try this is on Assault, although it will work on any game type), take the bomb and crouch walk at the bottom of the debris while looking just ahead of your own feet, this may take a few tries. Another player will need to drive the Hog back a bit and then gain a lot of speed in order to ram you (the bomb carrier). You will be propelled the length of Zanzibar and land on the roof of the base. Remain crouching or you will show up on the motion detector. Be aware of the fan: it is quite easy to smack into it and lose the advantage. If you make it onto the roof, wait for your team to covertly join you. From there you have the element of surprise and should be able to gain the upper hand as there is actually a hole in the roof through which your bomb carrier can enter the base through.
This can also be done with Capture the Flag or Assault on Coagulation using the rocks where the Shotgun can be found. This is less risky if the person in the Warthog has the highest level of handicap on as they will do less damage to the player being rammed.
Warthog jumping in Halo 3
On Forge in Halo 3, Sandtrap: Make a bed of plasma batteries (2 by 4) and place a Warthog on top of it. Build a wall of fusion coils or power cores in front of the Warthog and repeat the process behind it. Do this as high as the Warthog itself. It is your choice whether you put walls on the sides or not.
Spawn some Overshields next to the driver seat of the Warthog, and go into player mode. Hurl a grenade at the ground next to the explosives, then run into the Overshields and get into the Warthog. You will be launched into the air while doing while driving the Warthog, and usually be killed when it ends its descent, but it is possible to survive. This can also be performed on other vehicles such as the Mongoose, Type-25 Chopper, Type-32 Ghost, or Prowler.