The pneumatic tire is, by far, the most important development in the usefulness of bicycles. They'd be pretty bad to ride without the rubber and air supporting you and giving you a super-effective suspension.
Choose a tire to fit your wheel's ETRTO specifications first and foremost, after that you should probably get the widest tire that fits your frame. It's more comfortable, doesn't really slow you down, and gets you better grip.
Unless you're riding in mud or doing mountain-biking, tread does basically nothing on a tire. The threads in the casing and the rubber are what improve the feel and rolling resistance, as well as grip.
Some tires have particularly vigorous puncture protection: the Schwalbe Marathon line is famous for it. They're a bit heavier and don't have as springy and vivacious a ride feel, though.
If you're setting up your wheels tubeless, you do need tubeless-compatible tires. If you're using innertubes, you can use 'em too, you won't notice a difference.
