As a music teacher, a very common question I get asked is, "How do you make a sound on the trumpet?" A lot of people think that you literally buzz your lips in the mouthpiece or hum into the instrument or squeeze the lips together to make a high-pitched squealing sound. None of these is exactly right. In actuality, the air travels over your lips causing the lips to vibrate, which causes a sound wave to form inside the tubing of the instrument.
When you first get your brass instrument, the best thing to do is just spend some time experimenting with blowing air into the instrument. At first, you might not get any sound coming out of the instrument, but typically, with time, either a low C or middle G should sound.
If you're having trouble getting a sound to come out, here are a few easy steps you can take.
1) Purse your lips like you're saying the letter "M". This keeps the corners firm so that air does not escape from the sides of your mouth.
2) Stick the tip of the tongue through the lips, making a small opening between the lips. This is called your "aperture," or opening.
3) Bring your tongue back into your mouth. It should rest naturally with a slight arch in the middle.
4) Blow air through the aperture. Practice just blowing air through the aperture to start out.
This is how you make a basic trumpet embouchure.
Once you're comfortable blowing air through the aperture, place the mouthpiece on the lips and do the same thing, just blow air through aperture. This time it will be going into the mouthpiece instead of just out into the world, though.
Once you're comfortable blowing air into the mouthpiece, place the mouthpiece into the trumpet and blow. A sound should result. If not, keep at it; it will come with just a little bit of experience and experimentation.
So, to recap the basic principles of a trumpet embouchure:
1) The lips must be in the mouthpiece, roughly 50/50 upper lip/bottom lip.
2) The corners must be firm so that air doesn't escape from the sides of your mouth.
3) The lips inside the mouthpiece must be relaxed enough to be able to vibrate freely.