I've heard some talk about "finishing" or "break in" of the barrel of a new rifle;
Please explain....

Also does this go for all guns or just AR's?
Can this be done after trying out a gun, or will it not work?

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I'd just shoot it and keep it clean and don't worry about all the nonsense about break-in. Some people never even clean the bore until the accuracy starts to drop off. I am sure you will get differing opinions so mine is: break in is nonsense.

I agree with Woodstock. I just shoot and then clean. Nothing special.

Unless the rifle is some sort of benchrest-grade sooperultrasniper rifle that is expected to shoot 0.10 MOA, and you're going to reload the same case for every shot after neck-sizing only, using bullets checked for concentricity, weight-segregated by the hundredth of a grain, etc., etc., AND you're a 0.10 MOA rifleman, it really doesn't matter.
Do it if you want to, but for the overwhelming majority of us, it won't make a bit of difference.
It can be done on a barrel that has a few rounds through it, but is best done on a brand-spankin' new one. The basic concept is to burnish the bore smooth by shooting, while thoroughly cleaning between shots. It's a long and tedious process for a possible tiny increase in accuracy, somewhat less copper fouling, and a bit easier cleaning.

Thank you Pigbat, that really helps me understand why you would "break in " a barrel.
I may be anal about my accuracy but .10 is not enough to worry about, I'm happy with 1-2" groups at 100yard. I doubt I'll ever have the chance to shoot @ anything over 150yard!

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Breaking it in is only really necessary if you want to split hairs with your rifle at 500+ yards. Most of us shooters can't do that. There are elaborate break in procedures which basically call for one shot and complete clean, for the first 20 shots. You'll go through 1,000 rounds and two whole cleaning kits at some of the break in procedures I have read for Rem 700's. Some speculate this is just to sell more barrels

Shoot it and keep it clean-ish. There is also some data to support that a slightly-fouled barrel is more accurate than a freshly cleaned barrel.