It's possible to run a website from your home but there is a lot to consider..
- Does your ISP allow this? - Do you have enough upload bandwidth to deal with your expected load? - Do you have a static IP address (otherwise you can host your DNS elsewhere and use a Dynamic DNS program to update it)
So it's possible but unless it's something that's not that important, I wouldn't bother. You miss out on a lot of the standard features of dedicated hosting providers such as redundant power, multiple redundant uplinks and hardware support.
For the handful of hosted sites that I manage and a version of Roundcube that connects to my mail provider - I use a VPS. Gandi is good.
I have a business internet connection at home. Even though it's mostly for personal use, it's a couple of extra dollars a month (not much) and I get 2 static IP addresses, and I'm explicitly allowed to run servers.
It's an excellent learning experience. But if it's for business, pay the money for a VPS!