www.site.com vs site.com in general settings

On one site, I currently have the http://www.site.com listed in the wordpress url and site ur fields, but if someone hits my site and doesn’t type in the www. they get a database error.

On another site, I didn’t put in the www in the fields, and no matter what is put in (with or without the www) the site can be reached.

Can I change the first site by taking out the www in the wordpress url and site url fields and not mess everything up? Will this solve my problem of reaching the site without typing in the www or is there a different problem going on here.

-I really hoped this made sense.