The term "algorithm" may sound like a bad jazz ensemble from the era of leisure suits, but it's really at the heart of what makes your computer tick. In the case of dive computers, an algorithm is a mathematical model - basically a formula - that uses variables to predict the future based on a set of known experiences. How are these mathematical models made? In simplistic terms, people went diving and recorded which dive profiles did and did not cause decompression sickness. A model was then written to conform to the dive profiles that appeared safe.
This process was first undertaken by physiologist J.S. Haldane at the turn of the last century. Since then, Haldane's model has been advanced, modified and fiddled with ad nauseum. Work similar to Haldane's was done by Swiss cardiologist A.A. Buhlmann. In fact, the work of these two legendary scientists is still evident in the names of the algorithms used in today's computers.
These models break up the tissues of the human body according to theoretical "compartments" that absorb and release gasses (i.e., nitrogen loading and off-gassing) at different rates. A model may use anywhere from six to 16 compartments, and they will range from "slow" to "fast" and have specific times associated with them (a 120-minute compartment, for instance, would be "slow"). The model predicts how long it takes a diver to load up on nitrogen and then get rid of it in each compartment and what the resulting decompression schedule (if any) should be for a given dive profile.
In more recent years, the Haldane- and Buhlmann-based models have been joined by the Reduced Gradient Bubble Model (RGBM), developed by researcher and technical diver Bruce Wienke. It approaches decompression theory by looking at the mechanics of bubble formation in the body. It proposes that adding deeper, shorter decompression stops, along with other steps, can help limit the continued formation of gas bubbles in tissues upon ascent, thus leading to less bubble trouble.
So what does all of this mean to the average diver? Not much. The fact is that bad algorithms don't last in the marketplace and just aren't used. Typically (and this is a broad generalization), those computers based on Buhlmann or RGBM models seem to offer more conservative profiles, or at least the ability to program in a greater level of conservatism than other computers. They also seem to be geared a bit more toward divers who intend to slip into decompression once in a while. Algorithms with Haldane's name attached seem to be not quite as restrictive and geared more toward no-decompression diving. (There are exceptions!) Your best source of information on a specific algorithm is the manufacturer. It's also good to note that a computer is only as liberal or conservative as a diver makes it. In other words, if you want more conservatism, just surface before "dive-time remaining" hits zero. It's that simple. Also, "liberal" is not good or bad - it's just different from "conservative."