I need to put forms on a website with fields that can be filled on by the user, then completed forms submitted to an email address.

Forms created in JotForm can easily be added to web pages:

http://www.jotform.com/help/67-Which-Form-Embed-Code-Should-I-Use

Usually the iFrame code tends to work best:

http://www.jotform.com/help/148-Getting-the-Form-iFrame-Code

The submissions can be forwarded to your email address when they are received.  These are our guides on setting up email notifications.  The first is on setting up notifications sent to you:

http://www.jotform.com/help/25-Setting-Up-Email-Notifications

The second is how to setup notifications to your users:

http://www.jotform.com/help/26-Creating-a-Form-Autoresponder