I personally own pretty much every Neo Geo system, except I'm still missing a Neo Geo CDZ
But I'll explain what each console is.
Neo Geo AESThis is the home version they released in 1991, the carts are true arcade (but they changed it to limited continues) Common are Japanese models that can run you $200-$300, while a US model can run $400+, whats the difference? well AES read games in English or Japanese based the the system, not the games. A JP or NA cart in an US system will read in English, but a JP system will only read games in Japanese. There's also a 3rd option, a custom console that has cheat bios. They can run cheaper than a US model, but still be pricy. Collecting games for this system is not cheap, while some games you can find under $20, others will cost over $1000.
Neo Geo MVSThis is the original Arcade, The arcade has carts you can change out, but you can also buy a custom console (this is what I use) So you can ether buy a full size arcade that may need a lot of maintenance or a buy a console and play it on any HDTV. A basic Arcade you can find find for about $400-800. And a custom console usually will cost about $500, Custom console usually include all the cheat bios. Carts on the MVS are the same as the AES only major difference is you can continue as often as you like. MVS carts are cheaper and many games where never released on the AES. But they are uglier compared to the AES. lol
Neo Geo CDThere's 3 versions
The front loader, only released in Japan and only 10,000 of these where made.
The Top Loader, the most common of the CD systems (at least the JP version is) the US version is hard to find.
And the CDZ, this is the system that reads CDs faster.
The CD system is for the most part one of the cheapest way to play Neo Geo games. Most games cost under $20 but you have to put up with horrible long load times, even the CDZ isn't the fastest, A JP CD system (top loader) will run you $100-$200, and the CDZ can be $250-$350. you can find JP consoles that have region switches in them so you can change it from Japanese to English, (not all games this works on) The CD system is just like the AES it reads games based on the system.
Neo Geo X This is the handheld emulator system that came out last year, It's okay but like everyone here has talked about, it's not perfect.
The games are on Flash cards and its not perfect by any means, short battery life, games look bad on the stretched out screen, this is not the way to go to play these arcade classics.