No, your browser is not going to attempt to try the www version of the domain name. It sounds like you are missing a DNS entry for the domain name without the www. You should have an A Record pointing to the IP address of your server for both www and without www versions of your domain name. The www is actually a subdomain.
Some DNS (such as that provided by Namecheap) can be set to automatically redirect to another domain. So if someone entered your domain without the www, the DNS would issue a 301 Permanent Redirect and the browser would then fetch the www. Otherwise, you can have the server issue a 301 redirect to the www version in your server's configuration file or directory configuration file such as htaccess (if you are on shared hosting) if you are using Apache.
Do a DNS lookup and see if there is an A Record returned for your domain name without the www. I have seen a few times where there was no DNS record for the without www version and this may be the case in your situtation.