How do I make a user input array or list in python?

There is a built-in script tool that does exactly this: Multiple Ring Buffer

You can right-click it and go to edit to see the source Python script for its implementation.

If you want to roll your own (as a script tool), the key to the implementation is a multi-value parameter (of type double) for the distances, which at runtime is passed as a semicolon-delimited string argument, which you can then split up into a Python list object using str.split(), which you can then loop over using a for statement and create the buffers iteratively.

Take a look at the argparse module (which you can download separately if you're not using Python 2.7). Specifically the nargs keyword, which automatically puts your data into a list. For example:

import argparse

parser = argparse.ArgumentParser(description='Make some buffers')
parser.add_argument('buffers', metavar='distance', type=int, nargs='+',
                    help='Buffer distances')

args = parser.parse_args()
print "Number of buffers: %i" % len(args.buffers)