In audio, an "effect" is typically the result of some form of signal processing, which can be applied live or in post-production, that changes the "clean" or "dry" input to achieve a desired aesthetic quality.
Audio effects can generally be placed in two categories. Gain effects function primarily by manipulating the strength of some or all of the audio signal. Examples include equalization, overdrive/distortion, and compression/limiting/gating. Synthesis effects work by analyzing the input signal and generating some artificial signal that augments or replaces the input signal. Examples include chorusing, flanging, phasing, reverb, delay, looping, and in a broader sense, using the input signal as a controller for a full synthesizer device, such as one using MIDI.
Video effects are similar in theory, but instead of an audio signal, they work on an analog or digital video signal by changing values for some or all pixels in an image. Examples range from basic color, contrast, brightness, and hue controls, to transitions such as fade-ins and wipes, to more specialized effects such as solarization, color-replacement ("green screen"), false-color, trailing and overlay effects, and similar. In video, "effects" can also mean "special effects", generally referring to post-production processing that adds elements to a "base plate" in which they were not filmed; these typically include compositing, where one shot is inserted or overlaid onto another, and computer animation, where something that doesn't exist in real life or that can't happen in real life is inserted into a live scene.