If you have a typical consumber based internet connection (DSL, broadband), you were likely assigned one IP address. This "typical" Internet connection then requires that you have some type of modem/consumer based router running "NAT", or Network Address Translation. Basically, it maps the public IP address for all of your internal addresses. The beauty is that by default, the mapping occurs on outbound traffic, and not inbound unsolicited traffic meaning that someone from the Internet would not be able to get into your private network, unless you created port forwarding rules to map the public IP address directly to an internal IP/Port.
So if you want to gain access to your internal SQL service from the Internet, you need to create a port forwarding rule to allow traffic on port 1433 to this internal device, but that also allows everyone else to do the same. You can only lock it down to a specific computer on the Internet if your router allows the rule to includes specfic external IPs.
Or, you can allow all traffic from the Internet to this SQL box using the port fowarding and run a local firewalll on the SQL server and use the server's firewall application in conjuction with your SQL authentication to allow access.
If you are not that comfortable in this space, you may consider avoiding this altogether and go with a hosting provider.
Just keep in mind that if you have more than one instance installed on your DB Server, you'll need to enable SQL Browser and use the instance to connect.
Another thing to bare in mind is the login method. If you're going to use Windows Authentication you'll need to use a valid user account in the DB Server.
Example: If you are using ASP.NET hosted on IIS: