It is widely believed that most search engines now look at the domain name when they go to index a page. However nobody really knows if this is really true as the search engine algorithms are actually secrets, like FICO scores. Google and Yahoo also constantly morph their algorithms to avoid favoring URLS that are using black hat SEO techniques from soaring in the page rankings unfairly. However to be on the safe side you might want to consider that this is a rumor that is probably true.
Domain names can be of any length up to 67 characters. You don't have to settle for an obscure acronym for a domain name like SEOE.com when what you mean is searchengineoptimizationexpert.com.
Having said that, there appears to be some disagreement about whether a long or short domain name is better. Some argue that shorter domain names are easier for people to remember and easier for people to type in the search engine box. For instance learnSEO.com is a lot easier to remember than mastercourseforoptimation.com.Others argue that a longer domain name is usually easier on the human memory – especially if it is a catchier phrase. The catchier it is the more clients are more likely to remember the domain name and type it into the search engine box.
One reason to stay away from acronyms is that they can mean more than one thing. The acronym ACA is a good example – it stands for several things including the American Counseling Association, The Association of Canadian Archivists and the Association of Canadian Advertisers just to name a few.
It is probably a good idea to go for a shorter name as long as it is to the point and meaningful. If it is longer than two words then make sure it is memorable.