Choose a domain name which describes your products or services. Use the same name as the title of your website; hopefully a brand to remember.
Look at how the website structure should develop, by identifying keyword groups. The domain name and directories, should clearly indicate content. Directories for this site are design, search-engine and promotion. Note:- We chose asimplewebsite.co.uk as it contains the word website, is easily remembered and truly reflects our approach.
You may own a well known brand and want to use it. If not obtain a domain first, which mirrors your products or services. We use UK REG to check availability and to register. Your first choice may well be already taken, don't despair. If it took a lot of effort to establish the brand, explore buying it from the current owner or consider another extension .uk, .com, .net etc.
A domain should mirror your brand, products or services. Unfortunately short meaningful ones, usually have been taken.
Note the maximum length of a domain is 67 and the extension 4 characters, so it can be relevant and contain a keyword or phrase, which will improve the chance of high ranking. I would avoid an over long name, as these are hard to remember, prone to typing errors and individual page url's can appear to be cut short by some search engines.
Using a hyphen may get the short name you want. They provide no advantage to ranking as search engines recognise keywords within a domain. A disadvantage is verbal communication and possible typing errors sending users to a elsewhere. The same can be said of using plurals.
A large number of the directories which provide information to search engines, list alphabetically by website title. Search engines concentrate on content and ranking; but generally seem to very slightly favour domains which start at the beginning of the alphabet. I would only look choosing a domain starting with a digit or say the letter A if it added to the brand.
Often a balance between availability and the ideal extension for your market. Is your business very country specific or would it benefit from an international audience? If a company is based in the United Kingdom use a UK domain or .com if it's an international business, think .org for not for profit organisations. Also look at brand protection by securing names with other extensions.