Clarinets are wind instruments, which means they make sound when air (wind) is blown over or across something to cause vibrations. Air is therefore a large component of clarinet sound, and how the air is produced and used affects sound quality and tuning.
There is a certain amount of air pressure required just to get sound out of a clarinet, which varies from note to note. This amount of air pressure, however, is the bare minimum. More is required to produce a rich, full, well supported sound. How is it produced?
First you must inhale well. Breathing in through your mouth, think of filling your lungs all the way to your belly button, and keep your shoulders and neck muscles relaxed. You should feel yourself filling up like a balloon, in all directions outward around your torso. Once you have all that air, you need a way to control it as you exhale and blow it into your instrument. Engage the muscles around your lower torso/stomach—front, back, and sides—the muscles you use when holding the crunched part of a sit-up. When you blow into the clarinet, these muscles help you control your exhale so that it has the air pressure you need, and is also controlled enough so that you can play longer phrases and with great musicality (with practice).
To locate the proper muscles, play a few notes while seated, but with your legs stuck straight out in front of you—this will force those muscles to support your body so you don’t tip out of your seated position. You can also practice this way. Wall sits (making a seated position with your back against a wall, with no actual chair present) will also engage these muscles. With repetition, supporting your clarinet sound in this way will become second nature, and you will notice that your sound has a richness and a clear focus that makes it easier to play with others, and to play in tune.
There are many ways to describe how to produce proper air support to play the clarinet. Another excellent description that approaches the topic differently can be found here.
Tuning is directly affected by air support, in addition to reed and mouthpiece choice, and to a certain extent by your instrument itself (some instruments don’t play in tune with themselves, regardless of the equipment used with them). Playing with less-than-adequate support produces a flat, unfocused, often airy sound. This can also occur when your reed is too soft. A reed that is too hard, on the other hand, produces sharper pitch and a covered, pinched sound; this can also happen when there is tension in your neck and shoulders. A renewed focus on support from the core muscles, and watching yourself play in a mirror, can help you learn to relax those muscles and allow your sound to open up.
Other factors will also affect the sound you produce on your instrument, primarily choice of equipment and voicing. See this page for more information on how to produce a well-focused clarinet sound.