Commodore owned the rights to the Amiga's OS, which had a ROM-resident part named "Kickstart", the rest came on software disks and was called "Workbench" (also the name of the graphical "shell" the Amiga used). You generally should have both, but at minimum you need Kickstart ROMs to do anything [0]

When Commodore died, it sold rights to Amiga Inc, who then made these licensing deals:

* Cloanto got the right to sell Commodore's existing Kickstart ROM and Workbench (AmigaOS 3.1)

* Haage and Partner got the right to make a modern software-only version of Workbench that still ran on Kickstart 3.1 ROMs (AmigaOS 3.5, 3.9)

* Hyperion Entertainment got the right to make the future version of AmigaOS that also works on PPC processors (AmigaOS 4)... but they also released AmigaOS 3.1.4 and AmigaOS 3.2, which are incremental improvements over AmigaOS 3.1

At that point, Cloanto started worrying that people would buy Hyperion's AmigaOS 3.2 to boot emulators and they'd lose their market for selling legal 3.1 ROMs wrapped in Amiga Forever. There was a dispute between them and Hyperion... but there was also a dispute between Hyperion and Amiga Inc over AmigaOS 4, it's a complicated story.

How does this affect you? Well, you can use _any_ means you like to run an Amiga emulator.

* If you want to mainly play games, you can look for a variety of pre-built bundles of WHDLoad-installed games, or TOSEC disk images, and you can find frontends (like LaunchBox, Retropie) already set up to launch your choice of emulator (WinUAE, FS-UAE, Winfellow, UAE4ARM, Amiberry, etc.) and boot straight into those games. If you want to purchase a legal set of Kickstart ROMs, rather than just Google "amiga roms download", Cloanto will sell you Amiga Forever. You don't have to use their frontend if you don't like them, the ROM images you're buying (including the 1.3 ROMs for even older games) are lightly obfuscated but emulators can still use them independently of Amiga Forever.

* If you want to mainly run an Amiga desktop, you should consider getting any of the ROMs (3.1 or 3.2), and install the corresponding Workbench on a virtual hard disk... maybe even upgrade that to AmigaOS 3.9? It's very comfortable. You'll also want to ensure you install the Picasso96 graphics system, so that the Amiga can see your real machine's native screen resolutions. If you still like games, install WHDLoad and you can either use individual WHDLoad game installers to read disk images of games and put them on your HD, or you can find packs where people have already done that for you. You can then launch games from Workbench and drop back into Workbench when you press WHDLoad's quit key.

* As a further extension... if you want to see a totally tricked out Amiga that's full of alternative/replacement software, the sort of environment you'd have if you were a software junkie back in the 1990s and installed everything that looked cool on Aminet... then get AmiKit. Someone did that for you already. It's full of goodies but it's not "traditional", and personally I hate its alternative Workbench (ScalOS) with a passion, give me Workbench 3.9 any day...

Finally, if you do buy the A500 mini, it's ready for you to put more games on it via an SD card... it expects WHDLoad-installed games: https://amiga500mini.com/GetMoreGames

[0] WinUAE also bundles an AROS ROM; AROS is the Amiga Research OS, it's a portable implementation of the Amiga 3.1 APIs so should be able to run existing Amiga software... but games are tricksy and depend on various OS internals that AROS doesn't promise to repeat, so whether you can play games using AROS as the ROM depends on the game