Pygame: how to change background color

It will redraw as red the next time you update the display. Add pygame.display.update():

background = input("What color would you like?: ")
if background == "red":
    screen.fill(red)
    pygame.display.update()

Or, you could move the pygame.display.flip() to after you (conditionally) change the background color.

See also Difference between pygame.display.update and pygame.display.flip

Create a variable to store the current color :

currentColor = (255,255,255) # or 'white', since you created that value

background = input("What color would you like?: ")
if background == "red":
    currentColor = red # The current color is now red

in the loop:

while running:
    for i in pygame.event.get():
        if i.type == pygame.QUIT:
            running = False
            pygame.quit()

    screen.fill(currentColor) # Fill the screen with whatever the stored color is. 

    pygame.display.update() # Refresh the screen, needed whatever the color is, so don't remove this

So now, when you need to recolor the screen, just change currentColor to whatever you need, and the screen will automatically turn that color. Example :

if foo:
    currentColor = (145, 254, 222)
elif bar:
    currentColor = (215, 100, 91)

BTW, I think it is better to store color as a tuple instead of a list, like red = (255, 0, 0)

Also, you don't need pygame.display.update (or flip) anywhere else than in the loop. What this function does it just take the latest shape/value of every drawn item and pushes it to the screen, so you only need it as the last item in your loop, so it displays everything.

In fact, screen.fill(red) changes the color of the pixels in the Surface object screen. You need to update the display after changing the color.
Note, however, that you should only update the display once at the end of the application loop. Multiple updates to the display per frame cause flickering. See also Why is the PyGame animation is flickering.

backcolor = white
if background == "red":
    backcolor = red

running = True
while running:
    for i in pygame.event.get():
        if i.type == pygame.QUIT:
        running = False

    # clear background
    screen.fill(backcolor)

    # draw scene
    # [...]

    # update display
    pygame.display.flip() 

Explanation:

You are actually drawing on a Surface object. If you draw on the Surface associated to the PyGame display, this is not immediately visible in the display. The changes become visibel, when the display is updated with either pygame.display.update() or pygame.display.flip().

See pygame.display.flip():

This will update the contents of the entire display.

While pygame.display.flip() will update the contents of the entire display, pygame.display.update() allows updating only a portion of the screen to updated, instead of the entire area. pygame.display.update() is an optimized version of pygame.display.flip() for software displays, but doesn't work for hardware accelerated displays.