Forms / Applications

Sorry, I know it’s probably been asked 100 times before, but searching through the forum is mind boggling.

I need to do a basic site, where potential students can fill in a few different forms that can be stored into the database and/or emailed to the admin.

Am I searching for the right terminology “forms” (I don’t want a contact form) I want a number of different forms that I can customise etc.

Can anyone please recommend a popular plugin that will do what I want?

Thanks!