They are called booties.  They protect the dogs' paws from a couple of things.  First off is ice.  Snow and ice are abrasive so they need to protect their feet.  These dogs run over 100 miles a day, that's a lot of snow and ice.  They also keep snow balls from forming in between their pads.  You know how irritating it can be to have a pebble in your shoe or between your toes?  They are not really for keeping them warm.  Dogs have a higher temperature then humans do.  They also don't have that mechanism that we do where when we get cold, the blood vessels in our extremities constrict in order to keep our core warm.  Their hearts beat faster so they have faster circulation then we do.  One way dogs cool off when they are running and it is too warm is to walk in water but it doesn't really work the opposite. 

They booties are either made from fleece or cordura with a velcro tab to snug them up around the leg.  In both of our 1000 mile races, mushers are required to have a minimum of 8 booties per dog in their sleds at all time.  When they go through open water or overflow, then the mushers change the booties.  When they are climbing steep hills, they are removed so the dogs can use their claws. 

Here's a video from our other 1000 mile race but it shows them putting on booties.