A splash page, or splash screen, is the "welcome" page of a website. It is the first page a visitor sees when visiting a website, so it is an opportunity to make a fast first impression. Not all websites have a splash page; in fact, splash pages are decreasing in popularity because they tend to be a hindrance to a site rather than increase functionality.
A splash page is usually a way to introduce visitors to the site. The page may or may not be animated and play music, but there is usually a "welcome" or "enter here" graphic of some sort. For a designer, this is an easy way to showcase a particular piece of work. The visitor to the site will then be able to click on the image to enter the actual website.
Splash pages can serve actual functions as well. The page might detail the optimum browser requirements for viewing the website, or give you a choice of which version of a website you might want to view. There might be a text-only version, or a flash or non-flash version. A splash page might also allow you to choose your language and region, or provide a warning or disclaimer about the website. Important updates to the website might also be posted on a splash page where they can be easily viewed. Finally, a splash page might be a place for advertisements.
For all of these reasons, a splash page sounds like it can be very beneficial to your website. However, many site visitors are annoyed by splash pages, and will end up closing the browser or leaving the site if they can't immediately bypass the page to enter the actual website. Frequent visitors might not want to view the same information or the same animation every time they visit the site, so it is important to provide a way for visitors to skip any media that might automatically play and allow them to enter the website.
Another problem that might be encountered when using a splash page is that because there is virtually no content on the index page of your website, search engines might have a difficult time indexing the page. Search engines might then skip over your website entirely, which would obviously be a problem. If you do choose to use a splash page, it is important to make it clear how to enter the website -- visitors to your site do not want to guess, and in most cases will not spend much time trying to figure it out.