The first and most commonly asked question I get is: "Why buzz at all?" Here's a post I wrote which addresses the basic reasons why buzzing "works."
I've noticed a bunch of posts on the Trombone-L about mouthpiece size and difficult high range, with the 6 1/2 AL being mentioned as the culprit a few times. Although the 6 1/2 AL isn't a particularly large mouthpiece in the orchestral field, it is at about this size, and at about .525 bore trombone sizes as well, that the particular problems I wish to address in this article begin to appear for many players.
The real reason many people find the 6 1/2 AL (and other, larger mouthpieces with fairly large throats and backbores) difficult to play in high registers (or loud, for any length of time), is that they haven't developed their embouchures to the point where they're providing enough resistance AT THE LIPS, but rather are accustomed to using the horn and mouthpiece to produce that required resistance. (Some resistance is necessary, of course, or else the entire contents of your lungs would whoosh out at one time. That's why the call it an air STREAM.) Tighter, more resistant horns and mouthpieces don't require the same strength and balance of embouchure as less resistant setups.
As a beneficial side effect of buzzing (and this is VERY important, maybe more so than the embouchure work per se) it will enable you to find the NATURAL angle (or angles) at which you should hold the horn in different registers. Once you have a good, comfortable mouthpiece buzz, and WHENEVER you insert the mouthpiece into the horn while buzzing, try to insert the shank so that its sides go into the receiver exactly parallel to the walls of the receiver.
This often results in different horn angles on your face than the ones to which you're accustomed. Learn to hold your horn at the angle your physiognomy requires, if at all possible. The arms and hands are MUCH stronger and more adaptable than the facial muscles, and a slightly new horn angle, although initially foreign feeling, often produces startlingly good results.
The trombone is an asymmetrical instrument that changes its fulcrum every time you move the slide. Try not to let this fact dictate your embouchure. The only things that can prevent you from holding the horn at the angles that most please your chops are either arm and shoulder weakness (light, high repetition weight training will solve that in three easy weeks) or the occasional big necked player playing a horn with a narrow slide.
Some of this also addresses a couple of recent posts. Someone suggested using a practice mute to play in the sub-pedal range, and someone else was talking about the relative dangers and benefits of buzzing without the mouthpiece. Both extremely low practice (long tones) and buzzing (with or without the mouthpiece) can be used to find and develop a strong point of resistance AT THE LIPS. Once this has been developed, one can play larger and less resistant equipment, if so desired, without losing range or endurance. Using a mute to produce those sub-pedals is counter-productive to this end (as is mouthpiece buzzing with a finger partially stopping up the end, or using one of the artificial B.E.R.P. type resistance providers, in my view). It's certainly EASIER to do these things with artificial resistances, but the benefits are much less.
One way to look at almost ALL long tone/embouchure building/buzzing practice is to consider it an attempt to find and balance a strong and consistent resistance at the lips; not BEFORE the lips, in the throat and oral cavity; not AFTER the lips, in the mouthpiece and horn; but AT the lips, where it can be controlled and used while still having the freest flow of air possible.
We brass players are lucky enough to have an infinitely adjustable sound producer/resistor (synonyms, really), made of flesh, a tool only singers share with us. Reed players have to find a reed that suits their purposes that day; stringed instrument players must change their strings or bow to modify their sound. All we have to do is learn how to control our lips. The various buzzing techniques are particularly helpful in this process.
The next topic to be examined, once we have some idea WHY buzzing is valuable (there are plenty of other reasons, as well, but they'll be mentioned as we go on), is HOW to buzz. (What approaches are valuable and safe, what approaches can be counterproductive.) Let's deal with mouthpiece buzzing first.