Is Tai Chi a Martial art?
Tai chi is one of several soft-style martial arts also called "internal" martial arts. This name, "internal," is to distinguish it from so-called "external" martial arts, like Karate, where the emphasis is more on muscular force and not as much on developing chi. (More on this distinction is to be found on this Martial Arts thread - interesting chit chat there.) If you watch a Tai Chi practitioner, it is evident that the movements are blocks, kicks, strikes, and punches - things you would use in a fight. Therefore, Tai Chi is an art that is martial in nature.
So You Just Hope that People Attack You Very Slowly?
Har har har smart guy. Obviously, you hope that people don't attack you at all.
Here's the theory, which is cool and does work. In Tai Chi, you train slowly so that your muscles learn to do the movements in a way that is completely relaxed. I've written about this elsewhere, but Tai Chi does not work, neither for its health benefits, its meditation benefits, nor for its martial arts capabilities, if it's done with tension. When you go to apply your Tai Chi - in class as part of Push Hands, or wherever else - if you're not relaxed, your opponent will get the better of you. You train slowly to learn to relax, so you have a shot at relaxing when you have to go fast. You learn that the softer person has all the advantages in a physical encounter: the softer guy dodges better, turns force better, avoids strikes better, and never gives away his or her centre, so can't be moved as easily.
That Sounds Like Magical Thinking.
Yup, it does, but it isn't. It is a matter of simple physics. If you learn to do the movements of Tai Chi correctly (and this is no mean feat), you will understand how to use your whole body against an opponent. Your arm might be stronger than mine, but it probably isn't stronger than my whole body. In relaxing completely and using the body's natural structure, you are able to channel your opponent's force down to the ground, or turn it. Your arm might be stronger than mine, but that doesn't matter if you can't push my centre. Your arm might be stronger than mine, but it's not stronger than the floor, which you'll find yourself pushing against as you try to push me. That's the way it works.
But You Still Can't Use Tai Chi for Self-Defense.
You can, and I have. In more important ways, I use it all the time, since Tai Chi movement informs just about everything I do. It's a more efficient way to go about your daily tasks, to defend yourself against everyday injuries, as well as to fend off the occasional person who might do you harm.
However, there is one very important point here: just because you learn a martial art, and you can use its principles in class or as you're going about your day, doesn't mean you know how to fight. You still have to learn how if that's something you want to do. What you won't learn in class is how scary it can be when someone crazy or angry is being aggressive toward you, or how to stand up to that aggression (or avoid it altogether, which is preferable in most cases). Most people do not tend to cross the line over into physical aggression, but some do. Knowing when that's about to happen is a whole other deal.
Long story short, class can help you with discernment - being in touch with what's going on around you so you can work more skillfully with the people and situations you encounter in the rest of your life. Class can help you practice your skills with dealing with the physical force of another person. We practice this in the form of Push Hands, a partner exercise, and through resistance that your instructor gives you from time to time as you perform the movements, so you can feel how they really work. Slowly, you build confidence that the movements do work.
So Using Tai Chi for Self-Defense Takes a Long Time to Learn?
Yes! And some people never really learn to trust it. Here I'll finish with a long quote from a master. This is from Cheng Man-ch'ing's book Tai Chi. The topic is the secret to perfecting Tai Chi as martial art:
...there is a secret. But it is so simple as to be unbelievable. Its nature insists that you believe, that you have faith; otherwise you will fail. The secret is simply this: you must relax body and mind totally. [Sound familiar?] You must be prepared to accept defeat repeatedly and for a long period; you must "invest in loss" - otherwise you will never succeed. I succeeded to my present state because I pushed pride aside and believed my master's words. I relaxed my body and stilled my mind so that only ch'i, flowing at the command of my mind, remained. Initially, this brought many bruises and defeats. In fact, in some matches I was pushed so hard that I lost consciousness. But I persisted. I followed my teacher by listening to and heeding my ch'i. In crushing defeat, I forgot anxiety, pride, ego. by emptying myself I gave the full field to ch'i. Gradually my technique improved. Then, and then only, did my responses sharpen so that neutralizing and countering were the work of a moment.
That's Tai Chi as martial art.