Your domain name is the MOST important thing when it comes to hosting your website and email. If and when you buy a domain name, it is best to make sure that YOU are actually buying the domain name, NOT your web hosting company. This means that YOUR information is put into the Registrant, Administrative, Technical, and Billing contact information for the domain name.
This is very important because if your hosting company ever goes out of business, or you get into a dispute with them, you will not loose your domain name or access to it. Verification of ownership or to make changes to your domain is often done with the contact information you have listed, either via email, fax, or snail mail. If ANY of these methods are insecure, you risk someone being able to steal your domain name.
In the end, your domain name is the lifeblood of your online presence.