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.