How can I make the email show that it's coming from the user?

When a user fills out the form, I want the email address they input to show as the 'From' address when the email comes through (instead of showing JotForm). Is this possible?

You can only have it by using Mandrill's SMTP:

https://www.jotform.com/help/240-Set-Email-Field-as-the-Sender-Email-Address-for-Notification-Email-with-Mandrill

Mandrill is now an Addon for MailChimp and you cannot use it if you won't upgrade your MailChimp account.

I would suggest setting up the Reply-To Email if you don't want to pay for an additional service.