What forces are there against the motion of walking or running? When we walk or run, are there any forces that counteract the static friction between our feet and the ground? I can’t seem to think of any, but if not, why is it that we still need to push on the ground to walk or run, even at a constant velocity?
 A: Well, there's gravity, but I'm assuming you aren't referring to that.  I'm assuming you're referring to pushing against the ground horizontally.
When your foot is on the ground, static friction holds the foot in place.  However, the rest of the body can move, starting from the ankles and moving upwards.  We basically end up pushing the body away from the foot.
It is actually theoretically possible to walk without this.  Car wheels are an excellent example of moving without applying backwards force.  You can also consider someone on ice skates.  However, in normal "walking," the motion we undertake is a sort of fall-and-catch pattern.  Most people lose balance in the forward direction until their front foot collides with the ground. Then you raise up on that foot and try again.  You have to push horizontally against the ground to make up for this motion.
There are many arts which teach ways to move which isn't this fall-and-catch pattern.  But most people do it because it is awfully efficient (especially with shoes to cushion the blow of the heel strike)
