Youtube is a great resource, but there are plenty of poor examples of T'ai Chi out there because you don't have to be an expert to post a clip online - check the feedback before trusting any 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 well known for the 'Simplified' 24-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".
(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 for Arthritis' form.