Well, I had this really nice description all typed out for you and ready to submit and then my laptop rebooted when I installed a new service pack for Office.
Ok, I'll try this again...
You asked about a couple different things and I'll do my best to answer them but I'm kind of new to this myself but I've been reading like a madman lately trying to get up to speed.
Delay - Not sure what you're asking about here. Delay effects are effects that use delay to create the effect. Echo/Reverb are both delay effects. Delay is a setting used in Echo/reverb to determine the amount of delay between the dry sound (the unaltered/original sound) and the wet sound (the modified sound that's been processed by the effect).
Echo - Echo is a term used interchangably with Reverb. You'll find that some equipment uses the term echo and others use the term reverb. If this isn' the case someone please let me know but the reverb (on my Fender PD-250) and the echo (on my 3 tray JVC/RSQ) perform the same function.
Reverb - Ok, so they're the same. What are they? I'm sure you're familiar with the term echo. You yell in a canyon and you hear the sound waves coming back at you in regularly spaced intervals in ever decrasing levels. But what does it do for me? Let's look at sound played in a live venue. Typically, large venues have their own reverb. Reverberation is the result of sound waves bouncing around a room as they're played from a single source and the effect of the overall sound when listened to at a point in a room. If you're in the typical civic center at a concert you'll find that if the music abruptly stops you'll hear the sound bounce around - or reverberate through the room creating the appearance of being in a large listening space. Ok, fast forward to later and you're listening to the CD at home. You're not in a big space anymore, you're in a smaller space with lots of soft stuff that will absorb the sound waves (carpet, couch, drapes, etc). To recreate the reverb effect, they take the original sound and play it back slightly delayed at an ever decreasing level to recreate the effect of the larger listening space. This can be done locally with assorted effects (predefined effects are found on many home stereo components - they're marked 'Hall', 'Concert', etc.) or on the original recording (by recording the effects). Depending on the room and the listening acoutics you might find that a little reverb on the mic inputs can do wonders for a room that has very little in the way of natural reverb.
Compression - Ok, to understand compression we need to understand two other things. The first is clipping. Clipping is what occurs when the signal becomes distorted or too large to handle properly (resulting in distortion as well). The second is signal to noice ratio. This is the difference between the background 'noise' inherrent in all circuits (yes, even digital) and the signal of the 'wanted' sound - ie the mic or instrument inputs. Compressors sometimes provide both compression and cut out. Ok, think about your last night at karaoke. A singer comes up and singer their heart out but they're the Kenny Rogers type. Sort of quiet, easy on the ear, but require a little 'boost' to get good volume levels. No problem, turn up the mic input and all is great. Next, up comes Rammstein. Screaming into the mic he's belting out Du Hast. What happens? The mic is overdriven, distorts, that distortion is passed along to the amp/speakers and nasty things can happen very quickly. A compressor cuts the signal before that distortion occurs by applying 'compression' or a reduction in the overall signal levels to keep them within the normal operating levels. Cut out is the exact opposite - sort of. As the wanted signal approaches the noise level it will suddenly go silent. This prevents the ratio from becoming so small that the wanted signal becomes 'lost' in the background noise.
Ok, now the questions come back at you.... do you want to apply these effects in software or hardware? What do you have now?
Software effects can be reproduced by most of the better software packages with varying degrees of success. There are also dedicated hardware selections that perform these same actions. Occasionally, the effects (reverb, etc) are built into equipment you might already have (ie Mackie DFX12 and Soundcraft GigRac both have built in effects). Peavy (among others) make a dedicated effects processor to be used externally (check the Peavy web site). A number of manufacturers produce compressors for use with mics (where the problem is much more rampant than on music inputs).
If you want to learn more about music, mixing, effects, and the like I highly recommend the Paul White basic series. I bought basic Mixers, basic Mixing, basic Effects and Processing, and basic Live Sound. I learned much on the items you mention above as well as other effects like chorus. If you want a little more technical detail I recommend the Yamaha Sound Reinforcement Handbook. All are available on Amazon and through most book resellers.
BTW, if anyone finds anything wrong with what I posted here please let me know. I'm learning as fast as I can but if I stated something incorrectly or steered someone in the wrong direction let me know.
Edit - Ok, I took so long to post this Lonman beat me to it.... maybe I should learn to type faster...