Arnold Jacobs preferred that his students buzz simple melodies, primarily in the middle register, focusing on generating a strong buzzing sound (i.e., not an airy sound). His most famous example of a simple tune was Pop Goes the Weasel, but he also used the Theme from the Carnival of Venice, Stars and Stripes, Christmas Carols, bugle calls, etc. The list is really limited only by your imagination. The purpose of buzzing simple melodies is to teach the lips to instantly vibrate the pitch you are hearing in your head, similar to the way the vocal chords vibrate the correct pitch when we sing. By using simple tunes, you immediately know of you are buzzing the correct pitch.
The reason your notes sound better on the instrument after buzzing is that you are doing a better job of putting the right pitch into the cup of the mouthpiece. If you buzz the correct pitch, the instrument simply resonates it (like a megaphone) and adds color to the sound, assuming you have the right valve combination for the desired note. This is especially helpful when working on a piece of music that is difficult for you to play. If you can buzz it on the mouthpiece, then it is a simple matter of coordinating the lips and the fingers. The reason we miss or split notes is that we are not buzzing the correct pitch and due to the sympathetic resonance characteristics of a brass instrument, the note will not resonate, and you hear it as a crack or split, or simply as a note with poor tone quality.
In addition to buzzing simple tunes, there are a couple of books that have buzzing exercises. Alessandro Fossi has written a method book called To Buzz, which includes a CD of piano accompaniments. You buzz along with the CD and it really helps lock in the pitch and train your ear. While doing these, he recommends using several different devices, including a simple plastic tube that is 10-20 CM (4-8 inches) in length and an Inspiron (with its relatively long flexible hose) to add the resistance that Bloke discussed above. Here is a link to the introduction video on Mr. Fossi's Youtube page demonstrating a few of his buzzing studies.
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The Brass Gym, by Sam Pilafian and Patrick Sheridan, has a section at the beginning with some buzzing exercises, along with many great exercises to be played on the tuba. This book is worth the price just to be able to listen to Sam and Patrick playing the exercises in octaves. This is a book that any serious tuba player should have in their library.
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