The half square triangle unit is probably the most commonly used patch in a quilt block -- well it is second only to the square. It is made simply with two triangles, filling up half the square each -- hence the name. Take a quick look through this website, or in any quilting magazine, and you will start to notice that this half square triangle, regardless of its size, is present in more quilts than not.
Best of All
Free Quilt Block Pattern
The Best of All quilt block is a nine patch which will give you some experience in using cut pieces to make a difference arrangement of the patches. I'm referring to the patches adjacent to the centre square which use a square, some half square triangles and some quarter square triangles. These pieces are joined in a frequently used patch (and I don't think there is a specific name for it), but it is not a common patch. There are also some flying geese units thrown in for good measure. All in all this is a fun sewing experience.
The two sample quilts illustrated are both set on point with sashing strips, but each has a personality of its own.