If it's a Java web app, then the server part is a servlet. Given an email message sent from a client form, your server needs to send that text off as an email.
There's code in the JEE stack to do this, or you can specifically download JavaMail. This will allow your programs to act as mail clients.
Your MTA receives messages from your servlet and sends them to the users. So far so good.
But you also need a postbox, i.e. the equivalent of a mail in-box for your users. Postfix, QMail and others offer a basic "flat" mailbox model, where mail is simply stored until the client picks it up, and then (usually) deleted. Access is via POP3. IMAP offers a lot more organizational capability, i.e. the ability to specify hierarchies of nested mailboxes, to transfer mails between them and so on. You probably won't want to create a GUI front end to all that complexity, so I'd guess you don't really need an IMAP server. You do, however, want a relatively simple POP3 server to allow your servlet to access the mailboxes via TCP/IP. This is usually part of the "standard" email server packages.
To have your own domain known to the world, you need access to the MX records of your DNS service, i.e. you have to set up one or two of your hosts, on an Internet-facing address, to be your post office.
Finally, if you want to save yourself a lot of trouble, be very careful in configuring your MTA (SMTP server) such that there is no chance for it being used as an open relay. i.e. it should not be possible for your users to send mail to the outside world in general (or hackers will find a way to abuse your Web interface to do this), and mail from the Internet should not reach your users. Most importantly, there should be no way for mail from the Internet to be forwarded to someplace else in the Internet. Find an open relay testing service (they're free) on the 'net and get one to run a test on your configuration once you think you're done.
EDIT:
Looking at Thorbjorn's answer, I realized you probably don't want your users receiving their mail through your app; they probably already have email providers and accounts of their own. In that case, you don't need to worry about inbox capability or a POP3 server. You could consider offering full email services at your domain but that's a very thankless job and if you have any choice, leave that dirty work to GMail, Yahoo, Hotmail and their ilk. Whatever service you provide will never please your customers enough, and you'll be fighting spam and other crime every day.