Embouchure refers to the way that our mouth is shaped when we play our instrument. *Fun fact: embouchure comes from the french word "la bouche", which means mouth!*
Flutes
For flutes, it is often effective to think about your mouth being shaped as if you were spitting a watermelon seed. This ensures that your aperture (the hole in your mouth that the air will be blown through) will be pretty accurately shaped and sized for playing your flute. Another thing I like to think about is that I'm blowing over a bottle top. The mouthpiece on the head joint has a hole in it that you blow over, and the bottom edge of this hole will be placed right around the middle of your bottom lip (for most people, you should be able to feel this edge on your lip). Position the hole so that you are blowing right across it, not directly into it. It should also be centered on your mouth, not to the left or right sides. It may take a little bit of experimenting and adjusting as to the exact placement on your bottom lip (for some people it may be more towards the bottom of your lip, others towards the top), and you can experiment rolling your flute in and out (moving the hole a little away from you or a little closer) if you're not getting a sound. Remember, you need LOTS of air and a fairly small aperture!
Clarinets
Start by rolling your bottom lip slightly over your bottom teeth, just enough so that they're covered. Next, your top teeth will act as an anchor on the top of the mouthpiece (about 3/4 inch down). Your mouth will now close around the mouthpiece. Make sure that you pull your chin flat. The corners of your mouth should be firm and form a line, and you should take care not to bite down on the mouthpiece (no clenching or clamping your jaws) as this will not allow the reed to vibrate and will also create tension in your mouth.
Saxophones
Start by rolling your bottom lip slightly over your bottom teeth, just enough so that they're covered. Next, your top teeth will act as an anchor on the top of the mouthpiece (about 1 inch down). Your mouth will now close around the mouthpiece. Make sure that you pull your chin flat. The corners of your mouth should be firm and form a line, and you should take care not to bite down on the mouthpiece (no clenching or clamping your jaws) as this will not allow the reed to vibrate and will also create tension in your mouth.
Trumpets/French Horns/Trombones/Baritones
Start by going "mmm", creating a line with your mouth and your lips gently pressed together. This is the shape your mouth should make while you play. You want to avoid your corners going up, or "smiling" while you play. Your corners (of your mouth) will always remain firm, but the middle part of your lips will be soft and flexible. A small hole in the middle of your lips, or the "aperture", is where you will blow the air through, and you always want to think about a steady stream of air coming right from your lungs through the aperture; it shouldn't spread out into your cheeks when you blow out (this avoids puffing our cheeks). The mouthpiece goes right in the middle of your mouth. French horns, you will be slightly different with 2/3rds of your top lip in the mouthpiece and 1/3rd of your bottom lip (give or take; this will vary slightly student to student). Trumpets and trombones will be right in the middle.