Gmail is nearly as synonymous with email as Google is with search. The Gmail system is so widely used and has features that make its competitors sweat to keep up with, it's hard to say no to it. Google Apps is the Gmail system, and almost everything else that Google offers with it, but branded to you. It's the same features and still hosted by Google, but your domain name instead of theirs. It's your very own company, organization, or personal email, document sharing, and calendar center.
Hosting a website requires more than just the quick installs most sites promise if you really want to keep things secure that includes the email system. Most "self-served" email systems that run on the same hosting space as your website are subject to more vulnerabilities, offer less storage space, and can suffer from server downtime. Google Apps solves many of the issues that frustrate people the most with email and it's also the same familiar interface if you have a Gmail account already.
The Standard Edition Google Apps account lets you add up to 10 users, create email lists and groups, and offers over 7 Gigabytes of storage space per account. This is the free edition, so recurring costs are not an issue.
The Premier Edition Google Apps account includes unlimited user accounts (at $50 per year per user account) and each account includes 25 Gigabytes of storage. With the premier edition you get an uptime guarantee, Google provided support, and access to other features like Google sync capabilities for more mobile devices.
When Google Apps is used for accounts, email records are pointed to Google servers, so moving between hosts or changing website hosting plans can be done without any email interuption and without having to migrate all your email info. This safeguard alone makes using Google Apps worth it. There is also far better spam prevention through Google as it's dynamically updated versus the server software that only has safeguards from the time it was installed.