This particular join has been one of my favorites since I heard of it and I use it lots.
The only supplies you will need are as small a needle as possible with an eye that will still accommodate the yarn you are using, a pair of scissors (maybe), and the two ends of yarn you want to join.
Basically, you just loop or hook each yarn end with the other and thread each end of yarn into itself.
Easier done, than said, actually, which is why I am here to show it to you now.
I like to loop or hook them together from the beginning because of all the times I've forgotten this step and had to start over.
VERY annoying, yes.
Thread your needle with one end and insert the tip of the needle between the plies of the same piece of yarn fairly close to the place where the two pieces are hooked together.
I sort of wrap the yarn around after I divide a ply to keep them from separating too much.
So, insert and wrap, insert and wrap, for quite a distance down the piece of yarn.
(so you don't have to do this process again because it's not long enough yet)
Once you get a good bit of yarn spliced and scrunched up on the needle, pull the needle (and threaded yarn, of course) through the splices.
Apparently, I'm showing this process in slow excruciating detail.
My camera was giving me fits (and so were the children) and then my legs fell asleep, so I may have lost track of what in the world I was doing.
I was using the timer on the camera, so I could use both hands in these shots, so it took a ridiculous amount of time.
AND the brand new batteries (normal, not camera rechargeables) were already starting to die, even though I had just put them in when I started this process.
AND, at this point I turned everything around because the WIP was on the right side of the table, and the yarn was really short.
So, smooth the scrunched parts down and pull the yarn tight.
Now just do the same thing for the other side.
Thread the needle with the other end and insert it back into itself between two plies.
Do the splice and wrap thing again. It doesn't really LOOK like I wrap it, but I do.
Splice and wrap until you think you've gone too far, I like to try and fill up the whole needle.
Pull the needle through and smooth down the scrunchy stuff.
Then pull it tight.
In the end, the result looks something like this. You can see where the two join each other from the little tiny loop there in the middle of the backwards S.
If the ends are too long, you can trim them, but be super careful. I did this whole process last night and I was too tired to be using scissors, I guess, and I must have snipped one of the plies. I only discovered it after I knit to that point, just past my join, and the yarn was coming undone. In case you didn't realize it, that means I've shot this process twice. I had much better light today, than last night, so it's all for the best, even though I was REALLY in the mood to knit last night. (which is why I forged ahead in spite of bad lighting)
Anyway, hopefully this wasn't overly detailed and decently understandable. Let me know what you think or if you need more detail, I'll add/edit if you need me to do so.