Before I answer the question, I just want to caution you that cracking, snapping, and popping in your joints is not okay if it is accompanied by sharp pain - or any kind of pain, really. Ask your teacher for help if you ever experience any kind of pain during tai chi. If the popping is painless, then you are experiencing something totally normal and even great.
According to a rather old article published at Scientific American online, cracking sounds can relate to one of two causes: a release of gas from the fluid that naturally cushions your joints, or a snapping of tendons and ligaments back into their correct places as you move the joint. I tend to think that the snapping and crackling you get during tai chi is much more likely to be the latter rather than the former.
The thing to understand about your joints is that over time, the ligaments that connect bone to bone in your skeleton and the tendons that connect your muscles to your bones become tighter. A small degree of tightening over time can be the result of aging, but it is more usually related to sedentary behaviour. When you don't acheive a full range of motion in each joint on a regular basis, the ligaments and tendons naturally get shorter. Reason number one million that your body is a use it or lose it proposition.
To get an idea of what I'm talking about, take a look at this x-ray of a healthy human hand:
See the black spaces between the joints of the fingers? Those aren't empty: they are occupied by cartilage, which doesn't tend to show up on an x-ray. This spaciousness between the joints is what you want. With that cushion of cartilage in place, your bones won't rub on each other, and chances are the joint will remain healthy and pain-free.
When your tendons and ligaments get tight, they draw the two ends of the bones together. Eventually, what you get is compression of the joint. The cartilage thins, and you end up with rubbing between the ends of the bones.
This is an x-ray of the finger of a person with osteoarthritis:
The official word on osteoarthritis to date suggests that it is of mysterious origin, but I've been taught that the major cause is the drawing together of the joints because of tendon and ligament tightening.
When it comes to the spine, this tightening can be quite dramatic. Here's a normal cervical spine - aka neck. Look at the spaces between the vertebrae. Each of those spaces contains a cushion of cartilage that contains fluid to ease the impact of any jarring that occurs to the spine. These cushions are also called "discs."
You've probably heard of "herniated" or "slipped" discs. These occur when the spaces between the vertebrae become narrow, and the disc is squished out of place, or tears and the fluid leaks out. This is an MRI image of a herniated disc (source).
See that bulge? That's the edge of the disk squishing up against the spinal cord. Yikes, right? This kind of squishing occurs because the tendons and ligaments along the spine become tight over time. The medical community will tell you that this is a consequence of aging, but more properly it is a consequence of a lifetime of not moving enough.
The whole point here is that if you stretch the tendons and ligaments, you can increase the space between the joints. Tai chi is great for all your joints - I've used it to stay pain free despite some extremely serious joint injuries. It is especially wonderful for achieving increased range of motion through the hips and shoulders, and for stretching out the spine. I'd be willing to bet that the bone cracking you experience during tai chi is most intense in the shoulders, hips, back and neck.
That cracking is the tendons and ligaments popping back into place as you elongate your joints. It can also be the bones themselves popping back into correct alignment - the tendons and ligaments can pull them out of place when they are tight.
At first, these noises can be a bit disturbing. Over time, though, you'll come to look forward to them. Hey, people pay a lot of money to chiropractors to achieve the same effect! Once you get the hang of stretching in tai chi, you can put yourself back in alignment fairly reliably. If your back or neck feels out of whack, a quick tai chi set will usually remedy it.
Your joints will thank you.