You’ll use this one when you run out of thread in the middle of a textblock. The advantage is that one of the threads can be very very short, so you don’t need to leave a long tail in order to work the knot. (Don’t try to just use the world’s longest thread to get through the whole thing in one go if you have a textblock of any thickness. Thread longer than an arm’s length will be a pain to work with and take longer to pull through the book. If you can knot successfully, you’ll be much faster working with a reasonable amount of thread.) I find it helpful to show people with two different colors of thread, just so you can see which thread is going where more easily, and suggest you might do the same to practice.
As to whether knots should be on the outside (on the spine) or the inside (in the gutter) of the textblock, I’ve heard arguments for both. In the instance of the knot on the inside, the problem is that over time, the knot thicker than the rest of the thread lying in the gutter, and will squish into the paper. If you open an old book and search for knots inside, you’ll likely see a bit of an indent in the paper on either side where the knot has been sitting. You may also see a hole where the knot has worn through the paper; this is the downside. On the other hand, the knot on the spine also creates an extra bit of thickness in only a small spot, which can make it hard to get all of the sections perfectly aligned at the spine before rounding and backing, and that causes other problems. Pick your poison, then:
1. Make a loop with the new thread (white), towards the end of the length. In this one I made the long end go over the short end.
2. Pull the long end up through the bottom of the loop to make another loop. This is your ordinary slip knot.
3. Slip the end of the old thread (black) through the loop made in #2.
4. Pull the ends of the new thread slowly away from each other (let go of the old thread). As the loop becomes smaller, the old thread will be pulled down into the knot.
5. You should feel a small pop as the threads change shape as shown here.
6. Once they look like this, you should pull both ends of both threads away from each other evenly until the whole thing is tight.
7. Here’s how it looks in the end, although I didn’t pull it very tight so you can still see which threads are where. The second photo is just turning the thread a bit so you can see the knot from a different angle.
I usually thread the new thread onto the needle at this point and keep sewing before going back to trim the tails, only because I discovered that this keeps me from accidentally trimming the wrong ends off the knot, thereby being forced to swear and re-tie the knot with an even smaller bit to work with.
Here’s a video that someone else made which shows the thing in motion, which might make it more clear.