Python number guessing game

number = random.randint(0, 20)

You just told the user that the number is between one and twenty, but you use random.randint(0, 20). What if the result was 0? One disgruntled player. You should be using number = random.randint(1, 20).

prompt = "> "

You aren't hard-coding the value. That's very good. Since it is a constant, however, you should use ALLCAPS.

attempts = 5
while attempts > 0:

You are starting out at the high point and decrementing until you get to zero. That isn't really the number of attempts. That is how many attempts are left. I would define a constant for the number of attempts allowed, define attempts as 0, and have the while make sure that attempts is lower than that number. This becomes especially useful when printing how many attempts it took. As it is, your program displays the wrong number: the number of attempts that were left.

try:
    guess = int(raw_input(prompt))
except ValueError: 
    print "Only numbers are allowed. \n"
    break

It's good that you try to validate the input, but you should use continue instead of break. One invalid input shouldn't end the program.

print "Nice! You've guessed it right. The number was %d." % number

Using % for a format string isn't officialy deprecated, but it is recommended to use .format().

What if the user wants to play again? You make him re-run the script. I would use a while loop and a function:

import random

PROMPT = "> "
GUESSES = 5

def guess_number(start=1, end=20):
    print "I have a number in my mind that's between {}-{}.".format(start, end)
    number = random.randint(start, end)
    attempts = 0
    while attempts < GUESSES:
        print "What's your guess? "

        # Simple validation
        try:
            guess = int(raw_input(prompt))
        except ValueError: 
            print "Only numbers are allowed. \n"
            continue

        if guess > number:
            print "Go lower."
        elif guess < number:
            print "Go higher."
        else:
            print "Nice! You've guessed it right. The number was {:d}.".format(number)
            print "And you got it with {:d} attempts!".format(attempts + 1)
            break

        # Increment the attempts each time the guess isn't right.
        attempts += 1


        if attempts == GUESSES:
            print "You didn't guess it; the number was {:d}. You lose.".format(number)
            break
        # Print with the color red.
        print("\033[91m You have {} guesses left. \033[00m").format(attempts)

if __name__ == "__main__":
    print "Welcome to the Number Guesser game."

    again = True
    while again:
        print "I have a number in my mind that's between 1-20."
        guess_number()
        play_again = raw_input("Would you like to play again? ")
        while play_again.lower() not in "yn":
            play_again = raw_input("Invalid input. [y/n] ")
        again = (play_again.lower() == "y")

I would say to go with what zondo said, except that you should probably get rid of the extra check for attempts == GUESSES.

if attempts == GUESSES:
      print "...the number was {:d}. Youlose.".format(number)
      break

Your while loop is already checking that attempts is less than GUESSES. You can just put the ending print statement at the end of the guess_number function.

As it was mentioned, the function with this change will print "You lose" even if you've won. Instead of breaking out of the loop when the correct number is guessed, you can use return to make the function exit instead.

Also, zondo said you should use continue instead of break on invalid inputs for guesses. In case you didn't learn about the continue statement, what it will do is skip the remaining statements in the current iteration of a loop and move control back to the top of the loop.