Do you generally register your website domain name under your first and last name or under your business name?
What is the best way to do this?
I'm guessing your intent here is to have a business website, which is to be an assest of that business. If you ever sell your business, this will be part of the sale.
If you already have a company name, open it in that companies name, pay for it through its bank account. This makes sure there is no questions about ownership later on.
If you don't and you wait a few years to transfer it and your company is now worth a million dollars, you may have issues selling it from YOU to YOUR company for $1. Depending where you live you may have capital gains tax or you may devalue your entire company because the last trasaction indiciated it was only worth $1.
It depends what you want to do. If you have a business already set up and the domain name is for that business, then buy it under the business and claim it as a business expense.
If you are purchasing the domain name but don't have a business set up yet, you can buy it in your personal name and then transfer it to a business structure later if you want to.
If the domain/website is for your business it has to be in the name of the business. Easy to sell later on is definitely a plus point but I would prefer concentrating on today. Think about this which of the 2 seems a tech blog - TechCrunch.com or Arrington.com -- It will be difficult to create a brand for a product or service which is sold under your name.
The only reason one should do it in his name would be when the business is already running under his name. Naing it something else will dilute the existing goodwill.
And at the end of the day if you are in love with your name (who isnt) create names that include both you and your business. Ex HuffingtonPost.com
Remember it will be difficult to create a store chain called Travolta and fight with McDonalds, Naming it the Burger King still is an easier option.