From the ColecoVision game manual:

PANAMA JOE. That's what they call him 'round these parts: though no one knows his real name or where he comes from. But one thing's for sure. PANAMA JOE's a daredevil from the word "go!" No risk's too great if the reward's large enough. Winning. That's what's most important to him. And more times than not, that's exactly what he does. Because he's tough, clever - resourceful. And sometimes downright pig-headed! Though occasionally (ah-hum), he's been known to get in over his head. Luckily, he's got a knack for getting himself out of hot water as quickly as he gets into it! Let's hope MONTEZUMA'S REVENGE is one of those times.

Montezuma's Revenge was originally programmed by a 15-year-old named Robert Jaeger for the Atari 800. At a consumer electronics show in Chicago, Parker Brothers happened upon young Mr. Jaeger displaying his game and made him an offer. He licensed the game to Parker Brothers and it sold more than 600,000 copies from its releases on numerous platforms see list below. The title may have enjoyed more time in the spotlight if not for the Great Video Game Industry Crash of 1984.

It is painfully obvious from the manual excerpt that the main character is a fellow named Panama Joe. You control his movement and jumping to guide him through the underground ruins gathering gems. You must deal with various obstacles and enemies through the use of your wits, reflexes, and the occasional item.

There are five different items you can find: Swords for dealing with enemies, Amulets to make you temporarily invincible (not in the original Atari 800 version,) Keys of various colors for opening doors, Gems give you points which in turn gives you extra lives, and Torches are used to illuminate dark areas and last for an entire level. When touched, an item is placed into your inventory and will be used automatically as needed. Swords kill the next enemy you touch, Keys open the next door you touch, and Torches light up dark rooms when you enter. If your inventory is full, you can't pick up gems even though they aren't stored in your inventory.

There are only three types of enemies in the game. The most common are skulls, which will either roll on the floor or bounce around the room. By themselves, they are easily jumped over or run under as appropriate. Snakes will sit in a particular spot and would be rather lame enemies if not for the fact that swords do not work on snakes. Spiders are the trickiest of the lot since they will climb ladders. They seem to favor heading left at the top or bottom of a ladder and climbing a ladder when running along the ground. This behavior is somewhat random so these are only trends.

When you enter a room, you can see the entire layout so there are no off-screen surprises. Many people enjoyed the game for this reason since you could figure out your route without the danger of random traps. There are fire pits to jump, ladders and chains to climb, poles to slide down, and conveyor belts to run over, as well as other obstacles. Panama Joe will automatically grab onto a chain that he touches when jumping (for that exciting Indiana Jones action.) You cannot, however, jump onto or off of a ladder. They can only be traversed from top or bottom (and only from a little to the left of center.) The game featured realistic falling damage. If Joe fell more than his height, he landed smack on his head. The PC version featured him kicking his legs in agony after his head was crushed, a graphical feature missing from the ColecoVision port. Some later rooms are palette swaps or mirror images of earlier rooms usually with different enemies and items. Each section of play is followed by a bonus round consisting of a treasure room in which you grab as many gems as possible within the given time limit.

When one difficulty level is completed, you progress to the next which features faster and more numerous enemies, different rooms, and even more darkness. Eventually, the entire level is dark and you must progress using only your incredible memory.

Releases:
Atari 800

Utopia Software 1983
IBM PC
BCI Software 1984
Apple II
BCI Software 1984
Atari 2600
Parker Brothers 1984
Atari 5200
Parker Brothers 1984
ColecoVision
Parker Brothers 1984
Commodore 64
Parker Brothers 1984
Sega Master System
Parker Brothers 1989

The ROM can be found for ColecoVision, 2600, and Apple II as well as appropriate emulators. The original PC port can occasionally be found on nostalgic game collection CDs.

Some version-specific tidbits:
ColecoVision: Part Number 9460
SMS: One of only four third party titles for the SMS. (The other three were Where in the World is Carmen Sandiego, King's Quest, and Cyborg Hunter.) Does not work when using Genesis controllers on the SMS.

There was a remake for Game Boy and Game Boy Color published by Take 2 on May 30, 1999. The GB version contains many of the same elements such as the items and enemies. Many levels are also the same but your view is zoomed in to show detail so you can't view an entire room at once.

See also: Montezuma's Return.

Sources:
Playing on PC years ago, playing on ColecoVision emulator, http://www.gamechambers.com, http://www.mobygames.com, http://www.classicgaming.com
If you have any other info, feel free to /msg me.
Thanks to Master Villian for some info on torches and amulets.