There are so many ways to cast on for various types of knitting. This page is here to record some of the various ways.
The Chinese waitress cast-on is a stretchy, short-tail cast-on that is described in a post by Liat Gat of KnitFreedom. It is apparently very stretch and gives a lovely edge.
Figure 8 cast on: This cast on also uses two needles. The yarn is alternately wrapped around the two needles. You may want to used a smaller needle to cast on because this technique creates a loose row where the cast on is. It is important to knit into the back loop of the first row (both sides).
Judy’s Magic Cast on seems to be a combination of the long tail cast on and the Figure 8 cast on. There are many resources for this cast on including one by knitpurlhunter.com showing the cast on in-depth and another which shows the cast on being used for two at a time sock knotting. Cat Bordhi has an in-depth video tutorial too.
Turkish Cast on – this is an invisible, two sided cast on where yarn is wrapped around two needles and knitted on each side on the wrap. You make one half the number of wraps as you need stitches. You knit one end of each stitch, turn the work 180 degrees and then knit the other end of the stitches. It is great for toe-up socks, bags or any in-the-round project with a closed end (three sides closed). Here is a video by Drops Design showing the process. Another video by PlanetPurl can be found here. And yet another video by MsLynneAsh. Here is yet another video by Patty Joy Tutorials which shows the Turkish cast on (though she does not call it that) for Two at a Time socks.