There are two main types of MRI system - "high field" and "low field", each using a different type of magnet.

The Hallmarq standing equine MRI uses the low field type of magnet, installed so that the horse is scanned while standing up.  Watch a video showing how it works here ...

See another video of Standing MRI in progress, as broadcast on British national TV here ...

Anasager Dyrehospital in Denmark has produced its own videos on Standing MRI in Danish. Watch the MRI video here and click on their YouTube channel for more videos on X-Ray, ultrasound and other lameness diagnosis techniques.

High field magnets are of the more usual tubular shape so the horse has to be anesthetised to fit in the magnet.  See a video (Michigan State Veterinary Medical Hospital) here ...

And another (Vienna) here ...

One provider of high field MR moves the anesthetised horse into a trailer to be scanned. See a video here ...

Dr Kent Allen of Virginia Equine Imaging discusses several lameness diagnosis tools in his talk from May 2012 at TheHorse.com. The talk itself is over an hour, but you'll find the MRI case studies in the first 20 minutes.