Paul & Erik
You register your domain with a "Registrar". There are many of these, offering different rates and features. In return for an annual fee they keep track of the ownership records and are the place that points to your web hosting service. You register for a year at a time and can register for up to ten years if you like.
There was a big registrar failure (RegisterflY) in 2007 that cost a lot of people their domains. So you want to be sure your registration is done with a financially stable and reputable registration company. We have several we have used for years. They offer competitive prices, long term stability and user interfaces that facilitate management of our large domain portfolio.
An ongoing hazard is failure to re-register the domain when it comes up for renewal. The registrars all send renewal emails but sometimes they go to expired email addresses, get deleted by spam filters, etc. THere is a 30 period when they can be recovered, but then they expire. Lots of folks make good money pouncing on expired names and reselling them back to former owners.
Paul and Erik will always call you before your domain name(s) expires. We also help with finding and registering a good name with a reputable registrar.