A mail server can take many forms. Whatever it looks like, this is what it does. At pre-defined intervals, the mail server connects to the internet and collects all the mail sent to the domain since the last collection. It simultaneously sends out all the mail which has been sent to it from computers on it's local network. On the mail server is a list of mailboxes.
Incoming mail is sorted into the appropriate mailbox to await collection.
No more calling the ISP to add or remove mail accounts for one thing. With a mail server and a domain name one is free to set up as many mail addresses as needed at no extra cost. Another plus is that co-workers are able to send email to each other. However the crux of the biscuit is that sending and retrieving email happens quietly, quickly, and automatically in the background. The email users don't even need to know that they have a mail server, let alone what it does.