The literal translation of Tai Chi Chuan is “Supreme Ultimate Fist”, although I don’t feel the literal translation does much in the way of providing real insight into what Taiji really is or what you can accomplish by practicing it.
There are hundreds if not thousands of variations on Tai Chi out there today, and although many can provide some benefits, only a rare few have preserved all of the value and knowledge that the original forms included.
This is partly due to certain aspects of lineage and family forms being kept hidden, as well as teachers who have altered forms without a proper understanding of what the art form is all about.
You should understand that Tai Chi is actually a branch of Qigong, so be wary of teachers that teach Taijiquan, but claim to have little or no knowledge of Qi or Qigong. It is the Qigong aspect of Taiji that makes it move beyond something that is purely physical, to something that can calm the mind and bolster the spirit.
There should always be a mind body connection when practicing Tai Chi, and this is something that is sometimes lost in the practice. People are often too eager to simply learn the next movement before they have actually drawn any real benefit from the very first movement. It is really the mind power that is the most essential ingredient to all practice. If you can master that the rest will come naturally and effortlessly.
Ideally those students with a strong interest in Tai Chi will be able to attend some of the in-person training opportunities, since it is important to get feedback from an instructor who can watch you and make necessary corrections as you move.
Tai Chi can be practiced in a variety of ways depending on what your goal is during each individual practice session, but there should always be an intention present.
You can practice very slowly for instance, holding each posture for several minutes. Your intention during this style of practice may be to build internal energy as you connect to the Earth. You can practice with smooth springlike movements, which energizes the whole body and awakens the tendons and ligaments.
These are just two examples of ways to practice Tai Chi, and there are countless others. It is better to know one form well and practice it in a variety of ways then to bounce around between various forms never gaining a deeper understanding beyond the surface level.