What good is running for our health?
Running is the most accessible sport. And it is a very good cardiovascular excersize. 
Thats true.What equipment do I really need to successully start running?
Well, since running is basically just a type of gait where all feet are above the ground, I recommend a good running shoe. 
What other types of gait are there?
Well there is a walking gait.  That is where one foot touches the ground at all times. 
Is there an official sport for just walking?
Walking
Walking (also known as ambulation) is one of the main gaits of locomotion among legged animals. Walking is typically slower than running and other gaits. Walking is defined by an 'inverted pendulum' gait in which the body vaults over the stiff limb or limbs with each step. This applies regardless of the unusable number of limbs—even arthropods, with six, eight, or more limbs, walk.
Difference from running.
The word "walk" is descended from the Old English "wealcan"
Running
Running is a method of terrestrial locomotion allowing humans and other animals to move rapidly on foot. Running is a type of gait characterized by an aerial phase in which all feet are above the ground (though there are exceptions). This is in contrast to walking, where one foot is always in contact with the ground, the legs are kept mostly straight and the center of gravity vaults over the stance leg or legs in an inverted pendulum fashion. A characteristic feature of a running body from the viewpoint of spring