Apple - Is an unlocked phone and a jailbroken phone the same thing?

No, they're not the same thing.

Jailbreaking is a process where the iOS software and device firmware that Apple ships is modified to allow unauthorized/unsigned code to run.

Unlocking means that you can use a GSM iPhone with any GSM carrier, not locked to one carrier in exchange for a discount or an exclusive sales agreement.

Jailbreaking is the first step needed to unlock a phone. Jailbreaking allows additional root access to the device and allows for other app store mechanisms to load apps and code onto the phone.

Some utilities, like redsn0w, will automatically unlock certain models of iPhone in addition to jailbreaking.

Apple has sold authorized unlocked phones in several markets worldwide. As of June 2011, Apple will sell unlocked GSM phones in addition to the locked to Verizon and AT&T models. Some carriers will unlock an iPhone once your contract is up. There is a lot of confusion in the US on this point due to conflicting law of software licenses and Library of Congress exemptions that permit users to modify their phones for the express purpose of unlocking them when not bound by a contract to one carrier.

As the other answers say, unlocking and jailbreaking aren't the same thing.

There are two ways to unlock a locked phone: You either talk the owner of the lock (i.e., the phone company whose network the phone is locked to) into giving you the unlock code (something which, AFAIK, AT&T won't do under any circumstance), or you do without the unlock code. The latter method requires a jailbreak, since the unlock software certainly falls under the “not approved by Apple” umbrella.

(On a side note, in Europe, phone companies are required by law to give you the unlock code for an unlocked phone once your initial contract (for a year or two?) is completed.

iPhones have a built-in mechanism that ensures that all code running on it is under the control of Apple (this is called "code signing") and the "key" required to sign your code is one of the things Apple charge you for as an iPhone developer.

Jail-braking disables this mechanism, and allow you to run arbitrary code. This is potentially very dangerous, but may be required to do things not allowed by Apple. A great deal of effort is being done by Apple to stop jail-braking.

An unlocked phone is one not tied to a specific phone company, allowing you to subscribe at any phone company if you want to.