A 3rd-party mail hosting provider handles all aspects of your domain's email. You point your domain's MX records to their mail servers so that mail sent to your domain goes directl to their servers. They provide spam filtering and webmail. They also provide POP and IMAP access to your mail as well as SMTP servers that can be configured with local email clients like Outlook.
Wikipedia has a good comparison of webmail providers.
Google Apps
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Using Google Apps for your domain, every account at your domain will have access to the same Gmail webmail interface your users are used to with their "@gmail.com" accounts. However, all email sent and received will be for your domain name, not for "@gmail.com"
You can also use Google Groups as mailing lists for your domain.
Features: See Google Apps email features.
Pricing: Starts at $5/user/month. See Google Apps pricing.
Rackspace Mail
Pricing: Starts at $2/user/month. See Rackspace Mail pricing.
Office 365
Pricing: Starts at $5/user/month. See Office 365 pricing.
Zoho
Pricing: Starts at $0/user/month. See Zoho pricing.
FastMail.fm
Pricing: Starts at $4/user/month. See FastMail.fm pricing.
