It's hard to configure apache rewrite module for beginners, and it easy to get trouble. So use with caution.
There are two ways to configure the rewrite as I know. One is configure in httpd.conf
I prefer another simple way by add a .htaccess under the directory. In your case, it's www.mysite.com's web root
with following setting:
###################
RewriteEngine on
RewriteRule ([A-Za-z]+)[/]?$ artistpage.php?artist_name=$1 [L]
###################
This is the way it work, but you may need spend some time to try it out. I didn't test it