Tai-Chi as a martial art can be an enigma to a lot of westerners, because of it's obvious health benifits when it first appeared in the west it was touted more as an alternative health method than a form of self defence and it's martial origin's were often either downplayed or ignored. The most common reaction to Tai-Chi when first encountered is 'it's just moving slowly and poncy arm waving isn't it?'. This coupled with a few teachers of dubious merit making spurious claims of 'killer chi' and 'empty force' (knocking people over without touching them) has served to only further muddy the waters. Amusingly training outside of the UK this confusion is non-existent and the martial application of Tai-Chi is unquestioned.
It's true that taken at face value the slow movements of The Form (which are the core movements of Tai-Chi and what most equate with the whole art) do not seem martial in the slightest, but by actually practicing the moves and understanding the underlying mechanisms one can see what is actually happening.
Tai-Chi is primarily a principle based art, there is a great saying (which I can't for the life of me remember where it's from) 'Teach a man a technique and he has one technique, teach a man a principle and he has a thousand techniques'. The Form first and foremost teaches correct posture, efficient movement and an intrinsic strength from correct alignment (commonly called 'pun' in chinese) these are the cornerstones of Tai-Chi and everything else is built on them whether we are talking health or fighting. If your practice lacks this then it is empty and just waving the arms in the air.
To break it down in more detail. Tai-Chi is a close range art, it's strengths come from short range attacks, relaxed strength and heightened sensitivity to an opponents movements. Combined these are ideally used to over-power the attacker by gaining control of the their centre of balance and turning any energy used against them. My teacher Nigel has come up with 5 points that break down how Tai-Chi works in a combat situation.
The important thing to remember here is that once you get into range and gain control, you keep it. Not allowing your opponent to regain their composure or balance and using your skill of 'listening' to keep one step ahead of their movements. This is best summed up by the phrase 'the best arts are reactive and not active'.
If you instigate an attack then there is a conscious decision to make that attack, even if this is just a flicker on across the brain, there is still a lag between thinking of making a move and actually making it. However if you react, simply by instinct, then there is no conscious process so therefore no lag time, this method is summed up in the Tai-Chi classics as 'he moves first, I arrive first'.
Obviously you want to make sure that whatever your action is it's one relevant to the situation, this is where repetitive training comes in. By ingraining certain ways of moving into your synapses through training the Form and other exercises, then testing that movement under pressure in push-hands and sparring, you train your body to react instantaneously to an attack without the interruption of conscious thought.