Transfiguration includes the magic of animating inanimate objects
Sadly I don't have the books to draw quotes from, but here is what I have gathered.
I have no idea if the Harry Potter lexicon is seen as a valid source of information, but it lists animating objects as part of Transfiguration.
Transfiguration is magic which changes one object into another. It is possible to change inanimate objects into animate ones and vice versa. -HP Lexicon http://www.hp-lexicon.org/magic/transfiguration.html
Additionally, the HP Lexicon lists the petrification of Mrs. Norris as transfiguration, implying that depriving an animated being of its animation is also transfiguration (again no idea how canon that is).
From Pottermore and the books:
Professor McGonagall transfigures a life-size chess set as her piece of protection for the Sorcerer's Stone - Pottermore biography on McGonagall learned in "The Chessboard Chamber" instance
While my original thought had been that she transfigured it by making it larger, the text says she transfigured a life-size set, implying it was already big before she transfigured it (although I suspect she did in fact make it bigger). She could have taken a regular chess set and just made it bigger, but she still would have had to change it to make its own decisions, and to not be as chatty as Ron's set was.
Similar instances of animation occur when McGonagall summons the suits of armor to protect Hogwarts during the Battle of Hogwarts, and when Dumbledore animates the statues from the Fountain of Magical Brethren during the battle at the Ministry of Magic. One interesting bit of trivia about Dumbledore:
McGonagall sent an owl to Hogwarts, asking whether she might be considered for a teaching post. The owl returned within hours, offering her a job in the Transfiguration department under Head of Department, Albus Dumbledore. - Pottermore bio on McGonagall, unlocked content
So Dumbledore was the head of the Transfiguration department, and when in battle one of his first instincts is to animate statues to help him fight. While hardly conclusive proof, it makes sense that someone would rely on their specialties when they're in a rush.
Some have pointed out that the spell to animate the Hogwarts statues, Piertotum Locomotor seems to be a variant of the Locomotor spell which is a Charm. It could be that animation is in the middle between Transfiguration and Charms, but throughout the books we primarily see Transfiguration experts using these spells.
This ultimately means that McGonagall transfigured the chess set by animating it to play on its own.