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I have always thought of overdrive as like an extra gear, but from reading this forum I think that assumption may be wrong. So could someone explain to me the how, what, and why overdrive works/saves gas. I'm not very car savvy so simplifying it as much as possible would be appreciated.

It lowers engine rpm and that means less gas most of the time and less wear and tare on the motor most of the time

Overdrive is your 4th gear, so stay in it as much as possible for best FE . The reason is this is your highest gear which means lowest rpm which is desirable because low rpm generally results in higher fe . This thread is excellent on this topic, with charts to show optimal rpm verus load...
http://www.cleanmpg.com/forums/showthread.php?t=12356

Basic definition is a gear whose ratio (engine : output) is less than 1. This is then multiplied by the final drive ratio (usuall 3.x or 4.x), so the engine is turning about 3 times as fast as the wheels.

My civic's gear ratios, with a final drive ratio of 4.058:
1st: 3.250
2nd: 1.762
3rd: 1.172
4th: .909
5th: .702

You could, in a sense, call 4th and 5th both overdrives.

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In an automatic that has an overdrive button, it controls whether it will use top gear or not. In a 4-spd auto, turning ON the overdrive lockout (turns off overdrive) will keep it from using 4th, and only use gears 1-3. You don't want this.

Hmmm... maybe what you're really asking is what is a lock up torque converter? Try this edmunds article (http://www.edmunds.com/ownership/techcenter/articles/43836/article.html) that explains how it works.

Both the overdrive gear and the torque converter (if it's an automatic) work together to improve FE. The lower gearing of the OD allows for cruising at speed with lower rpms and less fuel used. The Locked torque converter ensure the maximum amount of energy is tranfered to the wheels. Most modern cars use a lock-up toque converter. As far as overdrive any modern car will have a low gear ratio for cruising anywhere between 0.8 and 0.5. Generally the sportier the car the higher the gear ratio. Manuals also tend to use a higher gear ratios.

Overdrive originally came about as a way to decrease engine RPM's in highway cruising while still allowing for good acceleration from a lower final drive ratio (generally around 4:1).

Many older cars with torquey engines didn't need over-drive as much as modern small displacement engines. For example, the BMW 5 spd manual has a 1:1 5th gear. No overdrive. Engines revs are kept in check by a fairly "tall" (numerically low) rear axle ratio. We used to have a '97 M3 that would get 30 mpg in hwy cruising mode.

One of our "toy" cars is a '79 Spitfire that I recently purchased. This car is fitted with a 2.0L inline 6 from the GT6 (the coupe version of the Spitfire) instead of the original 1.5L 4. The transmission is a 4 spd with an add-on Laycock J-Type overdrive, which were optional in these cars. However, the rear axle in the car is from a GT6 non-overdirve car. The difference is the axle ratio - 3.27:1 for the std cars vs. 3.89:1 for o/d cars. The end result for my car is in 4th gear and o/d engaged, I need to be going at least 65 mph or the engine bogs.

While the British Laycock company is long gone, add-on overdrives are still sold (advanceadapters.com is one source) and are common amoung 4x4 enthusiasts as well as some classic car and hot rod builders. 4x4's set-up for real off-road use will usually have very low axle gearing. To make the trucks somewhat driveable on the hwy without the engines screaming, an o/d is sometimes fitted. Likewise, hot-rodders will run low gears in order to get those big, fat Micky Thomson tires smoking, but add an o/d in order to cruise comfortably to and from the car shows.