Most of us are familiar with Web sites with different extensions (.com, .net, .edu, .co.uk, etc.), but what's really the difference between www.mydomain.com versus www.mydomain.net? Generally speaking, unless you are a .edu or .gov (which are reserved for accredited educational and government institutions), there really isn't too much benefit to having a .net over a .com. Search engines do weigh .edu and .gov extensions more heavily than .com, .net, etc., which means that a link to your site from one of these extensions is like gold. But as far as search optimization is concerned, for the most part it won't matter what extension you choose for your site. The major exception to this rule is related to local search.
Google's goal is to provide searchers with the most relevant results possible. To fulfill this goal, a part of Google's algorithm is dedicated to the physical location where the search originated. For instance, here in the US when you do a Google search the results usually have extensions like .com, .net, .edu, and so on. These are all extensions associated with the US. However, the same query performed in the UK will return a majority of .co.uk extensions instead of .com extensions. This is because when someone performs a query, one of the things Google looks at is the geographic location of that search and then returns results that are most relevant geographically. It's unlikely that someone looking for "pet stores" in the UK will want to fly all the way to the US to find one, so US-based extensions are usually not provided in the results.
Knowing about the localization of search results is important if your site offers products or services in many different countries. If you offer products or services in the UK but you're based in the US, host your site on a machine that is physically in the UK. When you develop content, keep that country's terminology in mind. Keyword research is critical to getting your international site to rank internationally. Spellings can differ greatly between countries and so can etymology. Though search results favor the country code of where the search originated, if a non-native site is optimized better than a native site, the non-native site can outrank the native site. This being said, it's entirely possible for a .co.uk site to rank in the US and vice versa.
Another benefit of having sites hosted in different countries is that you can't get penalized for duplicate content. Say you're a Southern California based company with a site hosted locally and you're looking to expand to the UK market. You can have the same content on both the US and the UK site without penalties for duplicate content only if the sites are hosted in different countries. This means that the UK site would have to physically be hosted in the UK.
In general, there isn't too much of an SEO benefit to having different site extensions. Sometimes sites will even buy versions of their domain name with different extensions and redirect them all to the main site to reduce competition and spammers. But, unless you're trying to reach an international market, site extensions have minimal impact on SEO.