Paleo means prehistoric, early, or primitive.
Paleo Running means running the way we were intended to run. The way we ran eons ago, before the Nike Air Sole or the Reebok Pump were invented to mess up our gaits. We ran barefoot (or near-barefoot) and with a gait that uses the forces of nature to our advantage (which also prevents injury).
How did we run before we tried to correct our pronation or supination with expensive shoes and corrective insoles? No one had to teach us running form and we didn't have to buy fancy contraptions to "correct" our bodies for running. We were built to run. Running is (used to be, anyway) an instinct.
In order to survive on the planet 30,000 years ago, early man (and woman) had to run (and run well!) to flee from predators, or to persistence hunt. It wasn't an option. As we evolved, our bodies adapted to our environment and our endurance running needs to improve our survivability. For example, we have been given better heat tolerance than non-endurance (but speedy) animals like predators(cheetahs and panthers) and prey (rabbits and antelope). We even grew springy tendons that our closest evolutionary relative, the chimpanzee, doesn't share. The chimp is not made to be a runner. It's a better tree-swinger. Ever wonder why it can be as difficult to stand in a line for two hours as it is to run for two hours? Standing in place requires work from all the the muscles designed to keep our tendons locked in one position. Our bodies were built to move in certain ways (which don't include standing for hours or sitting in front of a TV or computer)!
If paleo running is an instinct, does that mean I can just go out and do it?
Not anymore. Not if you've un-instinctivized your feet and your muscles and trained them to run with additional (possibly injurious) support. Shedding one's shoes for the first time, or reducing one's footwear to the most minimalist of covering and support will induce soreness (and probably injury).
First, you're no longer using that natural and effective running gait. You probably had that wonderful, natural gait when you were a very small child, before you started buying well-soled, thick, springy running shoes.
Watch a three-year old run. Maybe not the most graceful sight, but I'll bet she lands midfoot and "fall-runs" (leans forward from the ankles, hips forward, lifting her heels - taking advantage of gravity).
Now watch the average runner, clad in thick, bouncy running shoes. Nine times out of ten, that runner is heel-striking. Bang! Heel hits ground (body takes the shock of it). Push! Runner uses his small calf muscle to push his 165-pound frame forward. Nowhere is he taking advantage of gravity. Everywhere he is setting himself up for injury. Take those fancy shoes off and try to run barefoot or in minimalist shoes and that runner will likely be on the bench in no time.
Second, you've likely never used the muscles required to run barefoot or in minimalist shoes before. The cushioned shoes that have become every runner's requirement have caused us to lose the muscles required to run safely barefoot. In my case, reverting to the natural, instinctive (barefoot) method of running resulted in me actually creating my arch from where none had previously been. My feet are now stronger than they've ever been. And more effective at doing what they're instinctively designed to do.
Neo: "Why do my eyes hurt?"Morpheous: "Because you've never used them before."
It's like that with running. Learn to run using your body as it was intended to be used. Let your feet do what they have been designed to do. But do it safely. Transition slowly to minimalist shoes. Get a coach who knows barefoot/minimalist running and who will provide you with feedback (seeing yourself run on video is a very big help.)
Go paleo! The runner instinct in you will love you for it.