Almost all Tai Chi practiced nowadays is either the Yang or Wu style (although a few practice the Chen or Hao).
Tai Chi styles are generally done in two ways- long forms and the short forms. Long forms yield more benefits for the student, but require a longer amount of time to be set aside for them and have a much higher degree of difficulty to learn. While some quality is definitely sacrificed, short forms yield a large degree of benefits and are much easier and simpler to learn.
Of all the Tai Chi short forms, the most basic and well known is the Yang 24 move short form. This article does a fairly good job detailing the subject:
http://www.taichimaster.com/tai-chi/tai-chi-secret-how-to-choose-a-tai-chi-style/
If you want to have access to the best resource on the subject and the different forms of Tai Chi, their difficulty, who teaches them and what they can accomplish I would recommend getting the book Tai Chi Health for Life.
Also, if you want to get technical, the basic components of Tai Chi exist within Qigong, and the best form that exists within Qigong (which while simple contains most of the attributes and benefits one would get from practicing Tai Chi) is known as Cloud Hands. It's actually simple enough to be detailed within an instructional booklet (as opposed to a 5000 page tome for a Tai Chi form), with the book on it being called "Opening the Energy Gates of Your Body."
My own opinion is that the best way to learn Tai Chi is to first extensively master the energy components within the form from a simpler qigong form before beginning the Tai Chi one. If you're open to that, The Energy Gates method is the classic one that's been taught for the past few thousand years in China!
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