Someone who criticizes from the sidelines.
This comes from the annoying habit of some people of giving unwanted advice to vehicle drivers. It emerged in the USA in early 20th century, as motoring was becoming widespread. The first reference I can find to someone being called a 'back-seat driver' is from the Daily Kennebec Journal (Augusta, USA), May 1914:
"When New York pitcher Vernon Gomez retires as a smokeballer he wants to become a smoke eater. Here he gets a tryout as a back-seat driver on a hook and ladder truck at St. Petersburg..."
Throughout the 20th century U.S. fire departments commonly used large articulated ladder trucks, known as tillers. These had both front and rear-wheel steering to enable the long vehicles to turn in city streets. That's what Gomez is pictured steering here.
The link between that form of back seat driving and the present meaning of the phrase isn't explicit, and there's no particular reason to attach any negative sentiment to it. It's possible that the phrase originated that way, but I rather doubt it.
[My thanks to Sue Watkins, National Genealogical Society/Association of Professional Genealogists, for interpretation of the Kennebec Journal story].
The figurative and derogatory meaning of 'back-seat driver' is unambiguous in this from The Bismarck Tribune a few years later - December 1921:
"A back-seat driver is the pest who sits on the rear cushions of a motor car and tells the driver what to do. He issues a lot of instructions, gives a lot of advice, offers no end of criticism. And doesn't do a bit of work."
The expression must have been in common use, in the UK at least, by 1930, when P. G. Wodehouse used it, without any explanation of its meaning, in Very Good, Jeeves!:
Quite suddenly and unexpectedly, no one more surprised than myself, the car let out a faint gurgle like a sick moose and stopped in its tracks ... the back-seat drivers gave tongue. "What's the matter? What has happened?" I explained. "I'm not stopping. It's the car."
We no longer need back-seat drivers in cars to nag us if we take a wrong turning; we now have electronic devices for that. I wonder how long it will be before someone coins a negative term for them?
See other phrases that were coined in the USA.