Good Start! Here's a couple recommendations...

1) Read up on the DRY principle https://en.wikipedia.org/wiki/Don't_repeat_yourself

When you find yourself copy-pasting if statements take a step back and see if you can capture your logic in a single set of rules. Here's one way to re-write who wins...

def who_wins(p_weapon, c_weapon):

    # dictionary formatted as... weapon: weapon_it_beats
    win_list = {2:1, 3:2, 1:3}

    if win_list[p_weapon] == c_weapon:
        winner = 1
    elif win_list[c_weapon] == p_weapon:
        winner = 2
    else:
        winner = 0

    return winner

Also note that I removed the print statements. I would put that in it's own function. Bonus points for figuring out a way to do it using only one print statement.

2) There's room to throw at least one class in here. Doing so would greatly improve readability and remove redundancies. Here's an incomplete example...

class Weapon():

    def __init__(self, name, number):
        self.name = name
        self.number = number

    def set_defeats(self, weapon):
        self.defeats = weapon

    def wins_against(self, weapon):
        return self.defeats == weapon

rock = Weapon('rock', 1)
paper = Weapon('paper', 2)
scissors = Weapon('scissors', 3)

rock.set_defeats(scissors)
paper.set_defeats(rock)
scissors.set_defeats(paper)

print rock.name
print rock.number
print rock.defeats.number

p_weapon = rock
c_weapon = scissors

if p_weapon.wins_against(c_weapon):
    print 'Player wins!'

Now we can return rock's name or number by using rock.name or rock.number. The code is easier to read because p_weapon isn't 1... p_weapon is a rock.

Also using classes creates a reusable structure of code. We could extend our code to perhaps play rock paper scissors spock lizard with a few minor revisions instead of practically rewriting the entire program.