There are many ways to do this and many factors to consider.
Do you anticipate many users?
If not, you could simply contact your web host and request the capability to create more email accounts. Then, as the administrator, you'd have to manually create those accounts as people asked for them, and they'd go through their email client to check email.
Do you want the email service hosted on your web host, or are you okay with a third party hosting the service?
If you want it hosted on your own host, there is commercial software you can purchase to do this for you. For example, Alt-N Technologies (altn.com) used to provide such a product. Not sure if they still offer it, but it might be worth looking into, and searching for other, similar, products.
If you don't mind a third-party hosting the service, again, there are many options. Everyone.net is popular. Goggle does it too, and you get all the usual Gmail features. You can choose from free or paid options. In fact, I suppose many other webmail companies offer similar services; just ask around.