One of the hottest quilt designs this year is using half square triangles to create any number of patterns. HSTs are very simple to create, but the myraid number of methods can be confusing. My favorite is by Jenny Doan of Missouri Star.
The method is very simple. Take two squares of matching size, sew a 1/4 inch seam all the way around, and cut an x. But to give it to you in pictures:
Take two squares of matching size.
Lay them right sides together. This next part is optional: take a 1/4 inch ruler, and make a mark on each corner, both ways.
This is what each corner will look like once marked. This helps you know where to stop sewing.
Start sewing on one side, and go all the way to the mark. Pivot, and continue sewing, making sure not to go past the mark at each corner.
When you have finished sewing, lay your ruler corner to corner and cut. Then cut the opposite angle to get an X.
Press open, and be amazed - four perfect 3.125" half square triangles, without cutting an actual triangle.
These are being used in a Frozen inspired quilt, Fractured Snowflakes. Full tutorial on that quilt to come, but this is the bulk of that tutorial.