Bending notes on the saxophone usually involves starting the note at the correct pitch and bending it downwards, or starting at a lower pitch and bending it upwards. (This is the opposite of the guitar which bends a note upwards from pitch or downwards into pitch). The vibrato exercise starts off with note bending, so you can use this exercise to get started. This is a technique that takes quite a while to master. It involves slackening the jaw which increases the gap between the reed and tip of the mouthpiece. In order to maintain consistent airflow you need a strong diaphragm muscles to do this. The following examples are upward bends, note the various lengths of chromatic runs which start off the bend.
From the movie soundtrack Tucker
Once you have got used to the note bending in the vibrato exercise, you can adapt it to get some much larger bends than on semitone. You will probably find it easier on higher notes of the higher pitched saxophone, for example it is possible to bend a top C on alto down two whole tones, and you can bend a top E down a fifth.
As well as, or instead of, dropping your lip to bend the note, you use your oral cavity and tongue position to enhance the note bending. It is very difficult to describe what happens inside your mouth to create the bend, but many people find the easiest way to understand the technique is to start with just the mouthpiece. While blowing a note move your tongue away from the reed back towards your throat (but do not change the shape of your throat, it should be ope and allowing the air column through as usual). You may need to curl your tongue slightly. More or less by trial and error you should be able to find the position of your tongue at which the pitch changes. Once you can do this, it is just a matter of practice – try it on different notes and each time bend slowly and evenly down and then back up at the same rate.