Effects use different types of technologies to alter the source signal. Most effects will fall into one of three categories:
Analog effects alter signals by means of some physical change to the signal. This is often by manipulating the voltage of the source signal with various electronics (transistors, tubes, etc). The first effects units were analog. Many effects manufacturers are still making analog effects.
Digital effects use microchips containing programs and algorithms for manipulating a source signal. Digital technology translates sound into code (for example, binary ones and zeros). That code is then manipulated by an algorithm to create the desired effect.
Digital technology is more space efficient and cheaper to make. Many people feel that digital sounds colder and more sterile versus analog.
Modeling effects are a special type of digital technology. Modeling attempts to digitally emulate or recreate the sounds made by other equipment like a vintage amp or effect pedal. That means the model must interact in the same way the real world piece of equipment would interact. As you can imagine this involves a lot of complexity and is only now possible with modern day chips and software. This is still a very young technology. Line 6's Bass Pod is one example of modeling technology.
There are also hybrid technologies combining the best parts of both analog and digital technologies.