How do I restart a program based on user input?

Try this:

while True:
    # main program
    while True:
        answer = str(input('Run again? (y/n): '))
        if answer in ('y', 'n'):
            break
        print("invalid input.")
    if answer == 'y':
        continue
    else:
        print("Goodbye")
        break

The inner while loop loops until the input is either 'y' or 'n'. If the input is 'y', the while loop starts again (continue keyword skips the remaining code and goes straight to the next iteration). If the input is 'n', the program ends.

Using one while loop:

In [1]: start = 1
   ...: 
   ...: while True:
   ...:     if start != 1:        
   ...:         do_run = raw_input('Restart?  y/n:')
   ...:         if do_run == 'y':
   ...:             pass
   ...:         elif do_run == 'n':
   ...:             break
   ...:         else: 
   ...:             print 'Invalid input'
   ...:             continue
   ...: 
   ...:     print 'Doing stuff!!!'
   ...: 
   ...:     if start == 1:
   ...:         start = 0
   ...:         
Doing stuff!!!

Restart?  y/n:y
Doing stuff!!!

Restart?  y/n:f
Invalid input

Restart?  y/n:n

In [2]:

This line will unconditionally restart the running program from scratch:

os.execl(sys.executable, sys.executable, *sys.argv)

One of its advantage compared to the remaining suggestions so far is that the program itself will be read again.

This can be useful if, for example, you are modifying its code in another window.