Hey, it's been a while. Since I don't have a very large audience, I don't think I'll get too many complaints. Over the years, I've played only two or three systems: SAYC, 2/1 GF, and Precision. When I played Standard, my very patient partner and I would add convention after convention until we had almost a completely different system than SAYC. Same thing with Precision, only you get to create more with that system since not as many people write about it.
I have read about more systems than that, though, some of them ACBL-illegal (they're pretty strict on what you can and can't bid). Since there are too many systems to explain each one of them, I will list out and describe the basic archetypes of bidding systems of the world.
1. Natural - any system where all of the 1-level openings indicate one of your suits falls in this category. Examples include SAYC, 2/1 GF, Kaplan-Sheinwold, ACOL (played in Britain), and Bridge World Standard. Some non-traditional systems include Fantunes (a system played by Claudio Nunes and Fulvio Fantoni of Italy), Ambra, and ETM (everything that matters).
2. Strong Club - any system that incorporates hands of all shape with an unlimited range (typically 16 or more HCP) into the 1C opening. This frees the 2C opening (where all the strong hands are opened in natural systems) to represent hands you used to open 1C. Examples include Precision, Blue Team Club, Superprecision, Power Precision, Caroline Club, Nightmare Club, and Blue Grover.
3. Strong Diamond - like the Strong Club except all the stronger hands are opened with 1D and the hands that used to open 1D are put in another bid (typically divided between 1C, 1NT, and 2D). Examples include Burgay Diamond, Magic Diamond, Liu Hong's Green Card and Simplified Blue Card.
4. Relay - any system where after the opening bid, responder (or opener on the rebid) bids the next step to ask for more information and keeps doing this until the full shape of one of the hands is known. Typically used with a Strong Club system, examples include Viking Precision, Symmetric, Toad Club, and The Way Forward. These are not legal in most ACBL-sanctioned games. Your club director may allow it, though, at the club level.
5. Ambiguous Club - any system that uses the 1C opening to mean any strong hand OR another type of weaker hand. Examples include Polish Club, An Unassuming Club, Roman Club, Tangerine Club, and Swan Club. I always thought any of these would be extremely fun to play, and they're completely legal!
6. Red - any system that incorporates transfer openings at the 1-level. Examples include SCREAM (Strong Club Relay Excessively Accentuating Majors), Red Grover, Crimson Death, and Moscito (very popular). These are not legal in most ACBL-sanctioned games as well.
7. Forcing Pass - any system where hands that are normally opened are passed as an opening, and hands that are normally passed are opened at the 1-level. Now, this kind of system is 180 degrees from what most people play, and it is an interesting concept to me, but ACBL bans this kind of bidding completely. If you're interested, though, some of the systems include Suspensor, MAJOR, Superlambda, T-Rex, SPREAD, and Tres Boof.
8. Unique - these systems don't fit into any archetype as they having opening bids very different. One such system is called the Captain system where each of the opening bids describes, in steps, the general shape, then rebids clarify points or more on shape, completely artificial openings and responses. Another unique system is called Janus, where all of the opening bids mean 2 different things, and opener clarifies which on his rebid.
So, if you don't like the fact that you have to learn defenses to systems like Precision and its ilk, be thankful you don't have to deal with Red systems or Forcing Pass systems. If you're interested in any of the above systems, I would be happy to send you a copy of the notes on them. There are other systems out there, being created by mad-scientists the world over, and if you have an interesting system, send it to me. I love that stuff.