Youtube is a great resource, but you don't have to be an expert to post a clip online, so there are plenty of poor examples of T'ai Chi on it too - check the feedback before trusting any of the videos. Training with an Instructor is the best way to ensure you are performing the moves correctly and safely for you.
The "Lady in Pink" is a well known exponent of the 24 form, Yang-style 'Beijing Simplified' form (the most popular across the world)
(don't let her great flexibility put you off ~ she has trained for many years!)
This Shibashi 18 posture Taiji Qigong video is by Shan-Tung Hsu's Hong Kong "sister". A very popular exercise especially in South East Asia. (Chinese with English descriptions. 1st 20 mins = breakdown, 2nd 20mins = flowing movements)
Here is a 50 minute talk by Dr Paul Lam on his "Tai Chi for Health Program". Clearly a fit 60+ year old back in 2009 despite being an Arthritis sufferer in his teenage years! Founder of the 'Tai Chi form for Arthritis'