Gmail – Organizational-sensitive Gmail link

Yes - use Google Apps for Your Domain.

It's free for up to 50 users.

Free GAFYD (as provided by Tobbe below)

UPDATE This answer is no longer valid if you don't already have a Google Apps account. Google Apps is no longer free for new signups!! (Thanks to asalamon74 for the update)

Depending on your domain registrar (or maybe you have hosting/email?) you can forward an email address (eg. joe@example.com) to your gmail account. Gmail can then be configured to send mail as joe@example.com.

Or, if you actually have a hosted mailbox for joe@example.com then you can get gmail to download (POP3) mail from this mailbox.

Either way, you can use gmail to manage your email.

However, a slight caveat with the send-mail-as approach (by default at least) is that it is sent via gmail's servers (by default) and this is indicated to the recipient in the form of a Sender mail header (which might appear visually as something like "sent via joe@gmail.com" in the recipients email client). But otherwise it does work quite well.

If, however, you already have your own external (SMTP) mail server through which you can send email for your domain, but just wish to use Gmail to manage your email then you can use your own SMTP server to send email. This avoids the Sender header being set and is entirely transparent to the recipient. The recipient will be unaware you are using Gmail - which is now simply an email client.