In the old daze ...
We all shot JPEG or TIFF and there was generally a one-image-to-one-file relationship. Most people simply organized by date and all was well.
The some started using Photoshop (or Elements, GIMP, or any other pixel editor) to really tear up their images - think layers, etc. As Photoshop is a destructive editor, it became desirable to make two files for each image - a master that you did not edit and a copy (or version) that you did. Now things are getting complicated. Not only do you have to track the masters (say by date) but now you have all of these versions running around.
Then came RAW. Now you had one file, out of the camera, that you could not edit. Perhaps a few in Photoshop, and, in some cases, your "final" JPEG. (Bad idea IMO, but that is another story.) And you had to use a RAW Converter to get your images to the point that you could even "see" them.
So you used a workflow that went something like: Camera>Photo ingester/super browser>RAW converter>pixel editor>folders full of files.
The someone said, "Hey, I have got a LOT of photos, like two decades worth. I need all of the images of Sally in a blue dress at the red rocks that I took last year, but not this year." Arrrgh! A simple date system won't handle that. And did you want the RAW, the PSD/TIFF or perhaps the final web compressed JPEG?
This is were Aperture simply gets up and flys away from the competition. Although, to be fair, Lightroom isn't that bad. But in simplest terms, either one is light years ahead of Bridge or the Elements Organizer. And most photographers, as opposed to graphic artists, don't need anything more than Lightroom or Aperture. Especially as they allow you to do your heavy editing in the RAW decode step where you have the greatest possible control.
The biggest advantage of these products is that they allow you to manage IMAGES while they manage FILES. You can generally ignore file formats, etc. and simply produce (and delete after use) as many copies as you need in whatever format you may desire.
Given the current prices of Aperture, Lightroom, and don't forget iPhoto, most photographers would be silly, nay masochistic, not to use one of them. Especially if you would rather be taking photos than editing them.
Free advice, worth what you paid for it.