Passive subwoofers are called "passive" because they need to be powered by an external amplifier, in the same fashion as traditional loudspeakers. The important consideration here is that since subwoofers need more power to reproduce low frequency sounds, your amplifier or receiver needs to be able to output enough power to sustain bass effects reproduced by the subwoofer without draining the power supply in your receiver or amplifier. How much power depends on the requirements of the subwoofer speaker and the size of the room (and how much bass you can stomach!).
To solve the problem of inadequate power or other related characteristics that may be lacking in a receiver or amplifier, Powered Subwoofers (also referred to as Active Subwoofers) are utilized. This type of subwoofer is a self-contained speaker/amplifier configuration in which the characteristics of the amplifier and subwoofer are optimally matched and are both encased in the same enclosure.
As a side benefit, all a powered subwoofer needs is a single cable connection from a home theater receiver or surround sound preamp/processor line output (also referred to as a subwoofer preamp output). This arrangement takes a lot of the power load away from a receiver and allows the receiver's own amplifiers to power the mid-range and tweeter speakers more easily.
Which is Better - Passive or Powered?
All other things being equal, whether a subwoofer is passive or powered isn't the determining factor on how good the subwoofer is. However, Powered subwoofers are by far the most commonly used as they have their own built-in amplifiers and are not dependent on any amplifier limitations of another receiver or amplifier. This makes them very easy to use with today's home theater receivers. All home theater receivers come equipped with either one or two subwoofer pre-amp line outputs that are specifically designed to connect to a powered subwoofer.
On the other hand, what you need to do in order to use a passive subwoofer is purchase a dedicated subwoofer amplifier, which, in many cases could be more expensive than the passive subwoofer you have. In other words, in most cases it would be more cost effective to just buy a powered subwoofer in place of a Passive Subwoofer. If you choose this option the subwoofer pre-out from a Home Theater Receiver would connect to the external subwoofer amplifier's line-in connection, with the external amplifier's subwoofer speaker connection(s) going to the speaker connections on the passive subwoofer.
The only other connection option is that is available for a passive subwoofer is that if the passive subwoofer has in and out standard speaker connections, you could connect the left and right speaker connections on a receiver or amplifier to the passive subwoofer and then connect the left and right speaker output connections on the passive subwoofer to your main left and right front speakers (see photo).
What happens in this setup is that the subwoofer will "strip off" the low frequencies utilizing an internal crossover, which sends the midrange and high frequencies to the additional speakers connected to the subwoofer's speaker outputs.
This type of setup would eliminate the need for an extra external amplifier just for the passive subwoofer, but could put more strain on your receiver or amplifier because of the demands for low frequency sound output.
The Exception to the Subwoofer Connection Rules
It is also important to note that many subwoofers have both line input AND speaker connections (see example). If this is the case, the subwoofer is a Powered Subwoofer. However, in this example, it is a subwoofer that can accept signals from either an amplifier's speaker connections or an amplifier/home theater receiver subwoofer preamp output connection.
This means that if you have an older home theater receiver or amplifier that does not have a dedicated subwoofer pre-amp output connection, you can still use a powered subwoofer that does have standard speaker connections available.
For more tips connecting this type of subwoofer, check out the article: How To Connect a Subwoofer to a Receiver, Processor, or Amplifier by Gary Altunian, About.com Guide for Stereos.
When purchasing a subwoofer to use with your home theater, check to see if your home theater, AV, or surround sound receiver has a subwoofer preamp output (often times labeled Sub Pre-Out or Sub Out). If so, then you should use a powered subwoofer.
Also, if you just purchased a new home theater receiver, and have a left-over subwoofer that originally came with a home theater-in-a-box system, check to see if that subwoofer is actually a passive subwoofer. The giveaway is that it does not have a subwoofer line input and only has speaker connections. If this is the case, you will need to purchase either an additional amplifier to power the subwoofer, or, if the subwoofer has both speaker input and speaker output connections, you may be able to connect the subwoofer to the Left/Right main speaker outputs of the receiver and then connect your main left and right speakers to the speaker connection outputs of the passive subwoofer.