An Explanation of TM from TM.org
This is the official explanation of Transcendental Meditation from TM.org including a host of supposed benefits. It's certainly a bold claim to say that TM has been around for thousands of years and passed down in an unbroken chain to the present day, although there certainly have been similar forms taught in, for instance, the Bhagavad Gita. In my understanding, this is just a classic mantra meditation. The mantra is repeated as a focus of attention. When your thoughts wander away, you bring your attention back to the mantra, which ultimately stills the mind. Calling this a totally unique form of meditation ("automatic self-transcending") and charging a price to be taught is the biggest critique most people have about TM.
Maharishi on TM
This is Maharishi talking about TM in a very 60's kind of way.
More Maharishi. Here he talks more explicitly about the method.
This is a Buzzfeed video that shows four people on a 60 day TM experiment.
Maharishi and the Beatles
The Beatles were a big part of popularizing TM in the West. This is an extended video that includes interviews and written quotes from members of the Beatles speaking about their experience of TM.
TM continues to have a wide variety of celebrity endorsements. Here's Jerry.
Another celebrity, Oprah, talking about TM with a somewhat religious lens. You can see the variety of ways people understand TM, from a way to simply relax, to tapping into their own creativity, to experiencing God and finding direction through the practice.