Simply, the Domain Age of a website shows when it was first registered and also for how long that it has been active online for.
From an SEO perspective, Domain Age is important because an older search engine is seen as more trustworthy. The reason for this is that it gives the website a chance to show the search engines it has built up an amount of relevant and trustworthy inbound links over a period of time. Even though this is the case, newer websites will just have to focus on creating quality content and getting the quality inbound links.
Many people purchase domain names that have been around for a long period of time. The reason for this is that these people want to give themselves more search engine ranking power. This can work, however, if someone purchases a domain that is 5 or more years old, Google will only see it as this old if its pages have been indexed for this amount of time. Googles Matt Cutts believes there is very little difference between a site of 6 months and a site of 12 months. For this reason, many SEO companies do not like to work with brand new domains as it can be hard to get them ranking for their chosen keywords.