The border of a quilt often acts as a “frame” around the body. It can coordinate or contrast. There may be multiple borders. It (they) may use the same fabric(s) as the sashing, or different fabric(s). Each border may be a single fabric or it may be pieced (like the body). The border(s) may have a “square” seam or a mitered seam. When they have a square seam, the side borders are usually sewn on before the top and bottom borders.
To create the border pieces: Measure the side you are doing and add 2” or 3”. (Don’t measure the top and bottom length until you’ve added the side borders, because you have to include the width of the body plus the width of those borders!) Lay the border piece on the body piece so that the excess hangs off each end about the same length. Pin in place just as if you were pinning two smaller pieces of the body. No tugging, no stretching, no puckers. Sew and press away from the body. Then do the other side and the top and bottom the same way.
For multiple borders with square ends, do each border separately. DO NOT sew the inner and outer border strips together first. When measuring, add an inch or so on each end, and trim (square up) after sewing.
For multiple mitered borders, you can do each separately or you can sew the multiple strips together first and sew them to the body. When measuring mitered borders, measure the quilt body, then add 2 times the width of the border, then add an extra 4”. This is done for all four sides!
Using directional prints in the border(s): If the quilt body has a definite top and bottom orientation (e.g., animals arranged in rows), then the top border and the bottom border should face the same way. The side borders can face in OR can be cut lengthwise (so top to bottom as well); this may require more fabric! However, if you have designed well (or are lucky), you may have enough left over from the body and / or back to use lengthwise without having to buy more.
Re borders with corner squares: For these I do a hybrid approach: I put on the side borders as I described (cut long, sew, press, square/trim). For the top and bottom ones, I measure the width of the body, add 1/2" (for seam allowance) and cut that border piece. So let's say my border is 3" wide finished. I would cut my 4 corner squares at about 4". Sew each one to the ends of the borders. Then I can pin the borders in place and match up the side border seam with the top (bottom) border-corner seams (pressed in opposite directions of course). Sew and press AND then square and trim the excess off the corner pieces.
Look at various quilts in my quilt blog for examples of each!