3D converters come in 2 major flavors:
1: Convert a 2D signal for use on a 3D display
2: Convert a 2D signal for use on a 2D dispaly
For situation 1: Usually the box will have a passthrough mode (so for actual 3D content, you pass through and do not use the box, this means frame packed, SBS, OU, Checkerboard etc) you just let your 3D TV handle that if possible. If your TV cannot handle that mode (ie your TV does not handle SBS 3D) some of these boxes can convert SBS into Framepacked or checkerboard which hopefully your TV can handle. For what the box can do, you want to check on the specs on the box itself and also of cousre know what your TV can handle so you can know what output to look for. When not watching original 3D content (ie 2D content) the boxes utilize some technology that tries to give parallax seperation to the image. You see this built into many modern day 3D TVs as 2D-3D mode. This usually relies on rules like bottom fo the scren stuff is closer, bright stuff is closer, fast moving stuff is in front of non moving stuff etc. My epxerience with this is it's fun, can work well in some scenes but gets it wrong in some scenes (ie anytime there is a bright light in the background of an other wise dark scene like a street lam behind some people, it will seem to be in front of the people due to the 3D box). This is pretty much never going to be good as original 3D. Both shot in 3D and professionally converted 3D content will be more accurate and probably have a better 3D effect. However it does make the image interesting to look at and gives some pretty impressive shots sometimes. Again this will be dependanto n the format the box puts out and teh formats your TV accepts.
For situation 2 the box will take 3D formants (often including Frame packed, SBS, OU etc) and convert them to anaglyph 3D which involes wearing red/blue type plastic glasses (think 3D from the 80s). These work on any TV that can show a 2D picture and will provide 3D however often not very well seperated and with massive color distortion. This functionality may well be in a box that does convertion for 3D TV as well. However if you havea an actual 3D tv you woudl never want to use this mode as it's going to be inferior to Active or PAssive 3D.
As for isa ctive or passive better? It depends on your personal taste and cirucmstances. I posted a lot i nthe AVS forum thread linked above and while it seems copletely negative about passive, it's just to counter the untrue positive stuff thrown out about positive.
Passive has it's plusses and while I would do a lot of research and test both technologies out myself I would say this:
Pro: Usually the TV is cheaper, the glasses are cheaper, they don't use batteries and have no flicker. It's easy to setup a group viewing situation due to these factors and the 3D effect is quite good on them.
Cons: They are half the vertical resolution of Active (read that above thread for more talk on taht detail) and you must be roughly even with the display or else you will get cross talk (if the display is above or below you can angle it up or down but you must be looking at it direct on with your eyes roughly middle of the height of the screen). If your TV is small enough and or you are far enough away the image artifacts from this can be overlooked however bear in mind at that range you are essentially using a 1080p screen when you can't resolve even 720p resolution at that range.
Pro: Full 1080p resolution to each eye. Less of an issue with vertical viewing angles.
Cons: Glasses tend to dim things significantly (with LED not too much of a problem, with plasma and projectors a bit of an issue), glasses need batteries/charging and are more expensive than passive glasses (although samsung has some for $20 now). Tend to suffer from horizontal viewing angle issues and some people note the flicker of the lenses as a con. Some people also note that action scenes exhibit tearing occurs due to the nature of the technology.
My take away, if your screen is 47 inchers or smaller and you sit yoru average living room distance away from your TV (about 8-10 feet) passives resolution loss is probably not a big deal, but check it yourself. If you want some fun 3D sometimes but aren't really serious about it, passive is fine. But if you are a 3D nut, want to do a lot of 3D viewing and really care about resolution - you really need to check out active.
Either way, my advice is if you really care about 3D more than just a fun fad for a few minutes, buy a 3DTV, don't mess with converters. That is unless your current TV/Projector is a beast worth the trouble of keeping and somehow converting...