There were several incarnations of the Project Orion concept. The original idea, conceived in the 50s, was to use a single large conventional explosion to lift the craft into the air, then use nuclear explosions (provided by modified nuclear warheads) to lift to orbit and beyond. At the time, the US, UK, and USSR were routinely conducting above-ground nuclear tests, and fallout was considered an acceptable cost if the nukes were set off in remote enough areas. This version was eventually scrapped in 1963, due to a treaty banning above-ground nuclear tests.
The second major incarnation was a much smaller, lighter-weight Orion craft which was designed to be lifted into orbit by conventional rockets (probably the first stage of a Saturn V), then use Orion drive to break orbit and travel to Mars or the outer solar system. This made little headway because 1) the improvement over a pure-conventional rocket was more marginal, since most of the energy cost in interplanetary travel is simply reaching orbit, 2) in order to fit on a conventional rocket, the Orion craft needed to be scaled down heavily (becoming less efficient because a major feature of the original Orion concept was that it became more efficient as it scaled up to larger crafts) and needed to be made with light-weight construction techniques (also negating a major feature of the original concept, that it could be "built like a battleship" rather than having to scrimp every spare ounce out of the design), and 3) it became known during this time period that setting off nukes in low orbit would EMP entire continents.
The third major incarnation was the pure fusion variation mentioned upthread. Instead of using modified nuclear weapons, it would use a variation of the inertial confinement fusion reactor concept: lasers, electron beams, or ion beams would hit pellets of deuterium and tritium, heating and crushing them enough to undergo fusion. This never got past the drawing board, since inertial confinement fusion on anywhere near the scale needed (or even the scale necessary to make a viable commercial reactor) was and is decades of difficult and expensive research away.
There's two major sub-concepts in the pure fusion variation: a resurrection of the original Orion concept, using pure fusion as a way of avoiding the fallout problem, or on a craft constructed in orbit that would use the drive to accelerate to several percent of the speed of light for an interstellar flyby mission (most notably in the Project Daedalus and Project Longshot conceptual studies).