A snippet is the visual appearance of the organic search results on a search engine results page (SERP). Technically, the snippet is the part of the search result beneath the title and above the URL. You can control the snippet by re-writing your website’s metatags as you are conducting on-site SEO. You should think of your snippet as the equivalent to your ad text in organic SEO. In other words, your snippet exists in order to give people information about what to expect on your website, and therefore you snippet should somehow encourage the right type of people to click on your link and visit your website. As discussed in more detail elsewhere, your click-through-rate is extremely important, so therefore your snippets are very important.
Snippets are a bit complicated, but if you want more information, you should watch this video from Google exec Matt Cutts about how snippets are identified.