What is the thing that holds on the front wheel of a Giant Roam 300 called - and where can I get one?

You mean the fork? The two arms that come down from the handlebars and hold the wheel on either side of the hub?

Or the release mechanism with the little arm that tightens and loosens the axle?

May the forks be with you!

You're likely talking about the quick release skewer: https://www.sheldonbrown.com/skewers.html

That website shows you images of what they look like and also show you how to measure your hub and bike for a replacement. Of course, any local bike shop should be able to help you out as well; they'll have them in stock. If your Giant Roam is like 95% of the bikes out there, your front hub spacing is 100mm, and your local bike shop will almost certainly have a new skewer that will fit your bike.

Did you break the skewer?

If you are missing the skewer, make sure you ask the bike shop for a quick instruction on how to use it when you buy it.

I.E. Tighten it mostly down, then flip the lever to lock in place.