Wow. Lots of basic questions, which is just what this part of the forum is for. I'll give some partial answers, but a search of the site on each of those keywords will give you more info. Hopefully, others will chime in as well.
Dither: a mechanism for controlling the quality of recorded sound when conversion from one digital word format to another is necessary. Here's an excellent resource that was pointed out by Recording Engineer in another forum here (thanks for that, RE!). It's a good one, although a little advanced for the complete newbie. Just the same, stick with it: read it several times until it starts to make sense...
http://www.digido.com/ditheressay.html
Bottom line: when going from a longer word (like 24 bits) to a shorter word (like 16 bits for CD distribution), how you handle the conversion is very important to the final sound quality. Dither allows you to control the nature of the quantization noise that is inevitably caused by the conversion. Just lopping off the bottom 8 bits sounds like dogmeat, so you enter the realm of psychoacoustics and do dithering (which basically artificially supplies noise that you _like_, instead of just accepting the noise that you _don't_). This isn't a simple answer, because it is not a simple subject. Sorry about that.
Phantom power: A technique developed to supply power to condenser mics by shoving it straight up the same mic cable that brings your signal back down. This emliminates the need to have a separate power supply (or internal batteries) to power the electronics built into condenser mics. Dynamic mics like the Shure SM57/58 do not need power at all, phantom or otherwise: they are "powered" by the sound itself.
I/O: input/output. Gozintas and comesoutas.
Monitors, powered or otherwise: pretty much a necessity for doing the highest possible quality recordings. Monitor speakers aren't designed to "sound good", like home stereo speakers are: they are designed to be as accurate as possible, so that you can hear what you are actually doing, and so that your final product can sound good *on as many target systems as possible*. Home stereo speakers are often anything but accurate, which can lead to doing mixes that sound good on them, but have way too much bottom, or all the mids sucked out, or other generally unpleasant things when listened to in another environment.
Having said that: you can mix on anything, IF you take the time to learn exactly what your mixes sounds like in many different environments (essentially compensating for the speakers). Learning to translate what you hear on your mini system (or boom box, or Auratones, or NS10s, or whatever) into what will be heard on someone else's car stereo/walkman/home system is a true art. In any case, the more accurate your monitor speakers, the easier that art is to learn, because you have less to compensate for. IMNSHO, monitor speakers are worth the money: there are some very inexpensive ones that will nonetheless make it easier for you to hear everything in your mix (intentional and unintentional!). Is your
mini system good enough? Search me. Go listen to some good program amterial (like your favorite CD) through some monitors, and compare...
Hope that helps.