Well he explains it there better than I ever could (see post #6 on page 1 and #104 on page 3)
If you mix on generic speakers you are making your mix sound good on only those speakers because of their distinctive sound. Play it on different speakers and it probably wont be right. Studio monitors are supposed to have as little of their own sound as possible, so when you play it on any other systems aftewards it's not such a problem.
I have heard of people using speakers that aren't 'monitors' because they know them well. People say you're mixing in the dark with standard speakers but maybe if you were in the dark at your house where you lived for 20 years you would still find your way around without too many problems. It could depend on the speaker as well, of course pc speakers are a no no and some standard stereo speakers will be a lot worse for mixing than others.
At present I'm stuck with some old JBL stereo speakers where I live. I have had them a long time and probably done most of my music listening in the last 10 years on them. I know them really well and I know the room sound really well so maybe I'm not doing too badly. I have real studio monitors in London where I stay sometimes and I have never really gotten better results when I've mixed on them. Probably because my mixing has really sucked in the past

I do want to get some monitors here soon though, I know what I'm doing a lot more now.
It's funny you mentioned cheap ass headphones because recently I have been using a headset that is intended for skype etc. I think the bass is really boosted to help save your ears on poor quality phone calls. I used them once to start working quietly one night and for the first time my mix had too much treble

Normally it has too much bass, so I've been using them as a starting point to overcome my problem. Once the treble is there I can adjust the mix on my speakers, otherwise I don't notice the treble is lacking so easily. It's unconventional but it's been working for me.
There are other factors like time limits, and style of music. If you are working to a strict time limit you need to know you are getting good results straight away, but if you have more time you can listen in various places and get an idea of where you are going wrong. Recording a live band and mixing it will give you a lot more issues than using samples and soundbanks. If you have a simple bass sound and you cut the lows on ther other sounds you might just get away with it.
It's what I like about that thread, he is helping you to get the best out of your situation. Learning to not ruin your mix will help you get better results regardless of what speakers you have. It's all relative. A pro engineer could probably get better results on a poor system than a begginer in a top quality studio imo. The better your listening situation is the less problems you will have.