Source: http://www.sing-like-a-pro.com/twomethods.htm 

Here is the basic process of the classical method, in a nutshell:

1. You would take a breath using a controlled stream of air, to create an even sound.

2. The breath then passes through your windpipe where the tone is created, and is sent into your resonating cavities located in your head and face.

3. The resonating cavities will then amplify the tone before sending it out of your mouth.

The classical method believes that if the breath is not taken and controlled properly with the diaphragm and surrounding muscles, the tone will come out poorly.  

If you do take your breath properly along with proper support and tone placement, you will have a beautiful and even sounding voice, with a smooth passage into the high notes. It also teaches the importance of keeping an open throat while singing. 

An open throat means that your breath and tone can flow freely into your resonating chambers without interruption, making singing more comfortable. You can keep an open throat when singing by learning to lower your larynx, or voice box, like you are about to yawn. You simply sing with a slight feeling of a yawn in the back of your throat. 

Another important factor to think about when using this method is the importance of developing a consistent technique. This is done by learning to control your body mechanism, and can only be achieved with practice. 

Once you find what works for you, you can carry it to different styles of music, and pretty much sing anything.

The second main method of vocal instruction to learn from is what some people call the "vocal register method". 

This method is based upon the use of vocal registers. Vocal registers refer to the different co-ordinations your vocal cords take on when singing in different groups of pitches. In other words, a vocal register is defined as a group of notes close together using the same vocal cord structure. 

Registers are divided by bridges, or in the words of the classical method, the passagio. Each bridge your voice goes through as you get higher up in your range has a different vocal cord structure. 

It starts in the lowest register, where your vocal cords are touching about 1/3 of the way. The remaining length is free to vibrate. As you sing higher up into the next bridge, they come together another 3rd, and then another 3rd for the next register. 

Many singers experience a "break" while moving through their bridges. Maybe you've experienced this yourself. This method teaches the student to smooth out the bridges, without that sudden break into the next “voice”, or register, and by doing this, you can develop a larger range, excellent pitch control and flexibility in your voice. 

This is done by practicing very specific types of scales, using some very different sounds then traditional scales are sung with. People usually notice their voice beginning to improve after a few sessions, and can achieve very good results after a few weeks of regular practice. 

One really important thing to remember when using using techniques from either method is to make sure you don't have any tension in your throat. Tension is the number one voice-killer according to both methods, and you won't get very far with either of them if you sing with a tightened throat. 

Here is a run-down of the basic concepts of this method:

1. Learning about the vocal registers, and being aware of the different structures the vocal cords take while singing in them. 

2. Learning how to smoothly bring the vocal cords together, to get to the next register without breaking.

3. Understand how the sound resonates in each register. 

4. Maintaining your voice, by regularly practicing a "vocal workout" involving scales and exercises using very non-traditional sounds, designed especially to drop all tension, smooth out bridges and give you control over your voice.