The free OpenEmu is great for running all sorts of emulators on your Mac with a beautiful frontend, and I heartily recommend it for consoles and computers, but its MAME support is classed as ‘experimental’ – indeed, you need to download a separate build to get MAME supported at all. It works great for most games, but you need to compile your own binary to get sample support for games like Zaxxon, and you can’t get access to the DIP switch settings for games, or advanced configurations.
Before you try and use it, though, you need to install the SDL2 framework, download it here.
Open the DMG, and copy the SDL2.framework to /Library/Frameworks
Alternatively, you can install it in <Your home directory>/Library/Frameworks if your access privileges are not high enough.
Once that’s done, unzip the MAME release you downloaded, and copy it to your Applications folder.
Next, to make things easy, download a useful little application called M64, here. Unzip it, and put it in your new MAME folder. It’s just a double-clickable application that will launch MAME. The original source for this is here. I found this after I’d been running MAME from the command line for a while (if you want to do it this way, the easiest is to open a Terminal window, drag the mame64 icon to it, and press enter).
Put unzipped game ROMs in the roms folder within the mame folder, and unzipped samples in the samples folder.
You can use your mouse or cursor keys to navigate the menus once launched, and press TAB in-game to bring up the options screen. Remember to setup Input Options (General) or per game, to make sure you can use your chosen controller.
This is enough to get you up and running, and there are lots of resources on the web for using MAME beyond the basics.