[Gramidius]

If I post reviews on YouTube of different games or programs to the music of some author. Will I have to pay the author of the music? And how do I get around the copyright? And won’t monetization be taken away?

[Chuuii]

Usually the author "by default" is against it, because the copyright law says that using someone else's track without notifying the author can be punishable by law. If he personally lets you play music in the background, then fine.

[Gramidius]

How to be in this situation? Do I need to look for alternatives then? Maybe services where you can buy music without royalties? Where can I find copyright-free music for commercial and noncommercial use?

[Chuuii]

Such services are easy to find and it will be many times easier to use such music, yes, you can spend a small amount of money to subscribe and buy music, and you can use such music an unlimited number of times. Here you can find copyright free music.  A subscription to the service will cost about $10, now it's New Year's discounts and you can buy a subscription for $7 a month. The number of tracks on the service is more than 2500, which is a very large figure for such a cost of a monthly subscription.