WinUAE or FS-UAE (Linux). However, to run most games you’ll need kickstart ROM files (containing the Amiga’s operating system) which are still copyrighted and, hence, you have to know where to look for them. For the classic games you‘ll probably want Kickstart 1.3. For modern homebrews you‘ll want 3.X.
I imagine it varies. First of all, many games are only 1.x compatible, not 3.1 compatible. So for those, you'd need original ROMs anyway, and not just any, but specifically 1.3 likely.
Then, there is the question of WHDLoad. WHDLoad contains patches for most Amiga gams and makes them being able to be loaded from a file on a harddrive instead of being booted from. I believe it also makes games work on Amiga 1200 which otherwise would have only worked on Amiga 500 and so on and it makes game work on 3.x which only worked on 1.3 before.
Of course, the original ROMs are going to be more compatible, but I think it's wonderful that there is an open source reimplementation of both hardware and software of such an iconic and wonderful platforms such as the Amiga.
WHDLoad used to be commercial software, but now it seems to be a free download:
It’s not free, but I think the best, and easiest, way to get started is with Amiga Forever. If you can hang on a bit, they often have big Black Friday license sales.
Don't forget AdvJam where most games (it's a competition to make small point & click games in 2 weeks) are played and put on YouTube. To find these on YouTube, for entries in say 2018, AdvJam2018 should find a list of videos.
