You will need to have a primary server and a secondary server established.
When someone tries to access your site, it is first sent to their ISP's cache of sites. If it is not found, it sends a "lookup query" to other DNS servers which then send the correct IP address back to the original ISP server, allowing your visitor to find your site.
I believe that web.com will have to be the one to change the info on their servers- you should probably contact them.
Also, just as a word of caution- I know when we looked in to hosting our own site about a year ago, our ISP did not allow "always on" connections, even though we had broadband. Even though it seems as if you are always online, you aren't and they wouldn't allow people to host their own sites because of the high traffic. You may want to double check with Verizon as to whether or not they will let you do this.
Another point I'm sure you have thought about, but just in case, is the security issue. We pay our hosts a lot of money to make sure our info is safe and that no one can hack into our stuff. It may cost more, but it's worth it to me to not have to worry about it.
As well- you might think about any downtime. We pay our hosts to make sure our site is up 24-7 and they have the staff there at 3 am to keep us online. I personally don't want the responsibility of dealing with any website issues at 3 am 😃... just another thing to think about if you plan on having a lot of visitors to your site.
I think there are a lot of inexpensive alternatives to hosting your own site, unless of course that is something that is challenging and "fun" for you to do. Otherwise I wouldn't recommend it 😃 It will give you a lot more time to concentrate on making your site great.
Good luck to you- I hope you get it worked out!
-Elizabeth