First, you just need access to a computer with a Web browser and a connection to the Internet. Second, all your personalized settings and email are stored on the Web site, so they follow you whereever you go. Third, and most importantly, it is free. However, there are some disadvantages. Generally, you will get more spam email, because some Web email services sell their email addresses to spam advertisers.
Security is also an issue.
For example, a vulnerability was exposed in the way HotMail handles some email with JavaScript, causing some unauthorized access to accounts. The problem was quickly remedied by HotMail, but it shows that these services are not fool-proof. On the other hand, if you email messages are that important, you probably shouldn't be using one of these services. The possibilities are endless. Mail seem to be the most popular right now.