I registered my 2 with easily.co.uk (neither have a site associated with them, yet)
Easily's hosting is pay-as-you-go, i.e. you only pay for the bandwidth you use, ideal for a low traffic site like a family blog. Not really recommended for a high traffic, high bandwidth, commercial sites.
Other names include uk2.net, 1and1.co.uk, fasthosts.co.uk and easyspace.com (these are remembered from reading computer magazines and should not be viewed as a recommendation).
Some companies charge you for the domain but give you free basic hosting, others give you a free domain when you buy hosting from them. You're likely to get the best deal if you get both from the same firm.
If you are just having a personnal page, i.e. low traffic and non-commercial, then you've probably got hosting with your ISP anyway. You can then set your domain to either mask or forward to your site hosted by your ISP.
If you're after selling things or running a forum you'll need commercial hosting, as most ISPs won't allow you to run databases.