Deduced reckoning is figuring out where you are after starting from a known position, by using your speed, direction and time. (It's effectively integration of velocity, if you want the calculus/mathematical version).
At sea you would know your direction from a compass, speed by any one of various means, and important complications like the wind speed and direction would be estimated.
Odometry is really just doing the same thing on the ground, instead of at sea. Instead you count your "footsteps" as an estimate of your walking speed. A wheeled vehicle would instead count wheel revolutions, that's equivalent to counting footsteps.
Odometry isn't very accurate on its own as wheel slippage, carpet "springiness" and uneven floors can affect accuracy. A separate heading sensor can help with accurate headings at least, though.
For an old (1996!) but detailed account of odometry, I like Borenstein's "Where am I" technical report, which has a lot of outdated information but a still-good description of odometry and its inaccuracies. It's linked from this page:
(Beware it's quite a large download.)