Download MAME games, but first download an emulator to play MAME ROMS.
I've written in the past about emulation and OpenEmu, a multi-console emulator front end that uses multiple emulation cores to provide a unified experience to macOS (OS X) emulation.
OpenEmu also sports MAME Arcade emulation support but its a bit tucked away.
Step 1: Download OpenEmu Experimental
Go to openemu.org and click the swatch and download OpenEmu Experimental, and install it.
Step 2: Install the MAME core
Once installed launch OpenEmu, open preferences and select cores. Click the install button next to MAME.
Step 3: Install games
Games can be nabbed from the usual suspects like emuparadise, the legality of this is fairly grey. OpenEmu only supports zipped roms so if the rom comes in a .gz or .rar, be sure to decompress its contents and rezip it. Name it the exact name as before (but with the new extension of zip). OpenEmu's mame core is version 0.149.1 so its significantly behind the windows version (from 2013 to be exact). Some sites will list the earliest known support needed for the rom in question.
Notably, you'll want access to the dipswitches, hitting F2 repeatedly while the game is launching will take you to most ROM bios screens that enable modes like free play. Note, I had issues resuming games and had to force quit MAME resuming so you may want to hold off resuming games.
Optional: NeoGeo Emulation
Emulating NeoGeo games requires one extra step, you'll need to get the NeoGeo rom, the one I used is from emuparadise (just go there and search for Neo Geo Bios) or try a popular search engine. Drag the Neo Geo zip into OpenEmu. You'll see the NeoGeo rom in your game list but ignore it and double-click the games as you normally would.
Good luck with OpenEmu.
Other recommendations from this blog:
Running up MAME Arcade emulation with MAME/MESS - How to setup the most current version of MAME on MacOS. OpenEMU's MAME core is 4+ years old, there's quite a bit of newer arcade games supported in more recent versions.
Installing Provenance on iOS 11 with a Jailbreak - Provenance is an iOS multi-console emulator based off of OpenEMU, supporting conoles from the 8 bit and 16 bit eras.