If you need email for your dedicated server, there are essentially four ways of storing and accessing it. They are not all equal and are very different in the way they handle mail and deliver messages to you.
With this method, your email client on your computer contacts the mail server, authenticates, and downloads any new email messages. In most cases, it will then delete those message from the server, unless you tell it not to do so.
Unlike POP3, IMAP leaves messages on the server and accesses them remotely. It still relies on an email client, such as Mozilla Thunderbird or Microsoft Outlook. The difference is that the emails themselves are not actually stored on your computer. IMAP only downloads the headers and retrieves an email only when you click to open it. Because of this, you can access an IMAP server from multiple computers and see the same messages.
Locally hosted webmail is available with many web hosting control panels, and you can find others that are free to install and use. These usually rely on IMAP to deliver their mail. The only difference is that you access it via the web, using your normal IMAP login. The advantage is that you need no extra software. The disadvantage is that you still have to manage your mail server like you would with IMAP or POP3.
Cloud-hosted email accounts from companies like Google give you all of the benefits of free webmail accounts like Gmail, with the branding of your own domain name. The decisive benefit is that you can allow the cloud provider to handle all of your spam and mail server management. This disadvantage is that all of your data is hosted on their servers and is essentially at their disposal.
There is no single mail retrieval method that is perfect, but knowing the benefits and drawbacks of each should help you decide which is best for your server and your users.