There are many ways to play Game Boy games. From the original systems that sold millions to units to some independent projects made for playing Game Boy games on the big screen, there are plenty of ways to get some boys to play some games.
Emulation
Not much to say. Pick the most accurate and current emulator for greatest compatibility.
The Original Consoles
DMG-001
The Game Boy
The original Game Boy's iconic look has been a trendy design for hipsters and neckbeards to blow their load over. The DMG is a big beige brick with big purple-ish buttons, can only play original Game Boy games, and has NO back-light. The literal 2 bit puke green display doesn't help too much either. Requires 4 AA batteries, and isn't very pocket friendly. In all honestly probably the worst Game Boy console aside from aesthetics.
Atomic purple GBC
The Game Boy Color
The Game Boy Color was a step up in almost every way from the original console. Featuring a display that can show up to 32,000 colors (56 at the same time), a smaller size, and a multitude of console colors available it is only a marginal improvement use wise from the original DMG GB. A GBC only requires 2 AA batteries, and is reported (by Nintendo, at least) to have a battery life of 10 hours. The GBC was almost made out of necessity due to rising competition from Sega with their Game Gear console, which had a full color display and a freaking back-light. The GBC is backwards compatible with original GB games but also features many exclusives, usually indicated by a clear cartridge. Black cartridges USUALLY mean that a game can be played on an original GB but features improvements that take advantage of the GBC's hardware.
The Game Boy Advance
Purple Game Boy Advance
The Game Boy Advance. The last "big" revision to the Game Boy line. Ditching the vertical design of the all it's predecessors, the GBA features a D-Pad on the left side of the screen, an A and a B button on the right, as well as the Select and Start buttons and also has two bumper buttons on the top of the console. It's display can show up 32,768 colors when appropriate graphics modes are used by the developer. The display still doesn't feature a back or front-light, sadly. There are mods to add a front-light but every demo I've seen of it looks awful. The GBA can play every commercially available Game Boy, Game Boy Color, and of course Game Boy Advance games.
There's a clamshell model called GBA SP which has frontlight (model AGS-001) and another -- and rarer one -- with backlight (AGS-101). The shoulder buttons on these models suck, but it's probably the best way to play them on the actual hardware. I'd recommend getting an AGB model (above) with modded backlit screen.
Oh, you can also play GBA carts (not GB/C) on the original NDS and NDS lite. On NDSi, you can do that with twilightmenu++ with gbarunner2 app, but with some limitations. The 3DS has some GBA games released as 'ambassador games' but you can't actually buy them since they were given to people who bought a 3DS until early 2012. Relax, people have made tons of rom injects so you can play GBA on 3DS with custom firmware installed.