Effects processors help to provide additional "interesting" sonic characteristics that don't exist in the original source sound - usually to help the sound "feel" like it's bigger than it actually is.  Although each of these can be achieved using physical or electrical phenomenon, it's substantially more cost-effective to use a digital device that emulates the effects.  Many times, these processors will be included in the software on a digital sound board, or can be added to an analog sound system via an external processor.  It's also helpful to note that this is not an exhaustive list.

Reverb

Every acoustical space has a natural reverberation characteristic influenced by its shape and size, and by the materials present (the hardness of the surfaces, the presence of objects to absorb and deflect the sound - like furniture or people).  As sound energy travels through a specific room, the energy will reflect off of certain surfaces and become absorbed into other surfaces - similar to the way that light will reflect off of a mirror, or will become absorbed by a curtain to darken a room.  The resulting reflections of sound energy are commonly called reverberation.

The human brain naturally utilizes reverberation to process the proximity of a sound source.  Sounds that come from a far away source will have a large amount of reverberation, where sounds that come from right next to the listener will have little to no reverberation added.  The typical result is that although the human ear will be able to sense both of these sounds, the sound with little to no reverberation added will much more readily tend to capture the attention of the listener.

The reverb processor, as a tool, allows the audio technician to figuratively place a sound in a different (larger) space - creating a larger sound from one source than the natural reverb would provide, or emulating a physical separation between two sources by affecting one signal, and leaving the other one dry (un-affected).

Different types of reverb processors will result in different characteristic sounds.

Whether the sound is actually processed through these mechanisms, or the effect is simply emulated, the end result is typically similar enough for the cost-benefits of emulation to be worth the exchange in quality.

Plate reverb: Sound is played into and resonates within a metal plate. A piezoelectric pickup captures the resulting reverberations.

Room / Stage / Hall Reverb: Sound is played into different types of rooms and then captured by a microphone at the other end, with different resulting resonance qualities, and different timbres based on the type of room.

Spring reverb: Similar to plate reverbs, sound is transduced into and resonates within one or several metal springs.  A transducer at the other end of the spring(s) captures the resulting reverberation.

Delay

In its simplest form, a delay processor will capture a sample of sound and then play it back after a set period of time.  More complicated delay processors will often feature the ability for the delay sound to feed back into the input, so that the processed signal becomes processed again at a proportion equal to the feedback amount (or %).  Many will also include tempo settings (or a tap tempo mechanism) to allow the technician to quickly establish a delay consistent with music without needing to convert a BPM tempo to time (typically milliseconds).

Some sound systems - particularly in large spaces, or outdoors - are designed with multiple speaker sets placed throughout the listening area.  In order to properly align the timing of the sound and prevent phase-cancellation issues, it's common to use a delay to offset the timing of the signal from speakers that are farther away from the stage.  In this instance, the feedback option is often set to 0, and the timing is calculated using the distance of the speakers from the stage and the speed of sound.

Chorus

The chorus effect is a specific type of delay effect, where the sound is reproduced in relatively quick succession, and where the delay time is varied with a cyclical Low Frequency Oscillator (LFO).  The resulting effect is essentially a subtle de-tuning and layering effect that allows the sound to fill more sonic spectrum than the un-affected signal would.

Flanger

The flanger effect emulates the effect of a delay where the playback mechanism creating the affected signal will the speed-consistency of the playback will subtly drift in its speed-consistency.  The summed combination of the affected (wet) signal with the un-affected (dry) signal creates a much larger and "fatter" sounding output that has some interesting tonal characteristics because of the variance in phase between the two signals as they're added together.