I am looking for a tool to catalog images (and other media) that would let me retrieve an image (for insertion into content) based on several search criteria.
I would strongly warn against using the Media module. And this is from a person who has spent many years implementing the Media module on many major sites. It still has virtually-insurmountable problems with its WYSIWYG integration along with lesser-but-still-significant bugs of other types. I don't want to take away from the many great developers who have contributed to it over the years, but I just cannot in good conscience recommend it as it stands.
I would, instead, point you to the Scald module. Its approach of having a separate media entity (called an Atom) is less intrusive than Media/File Entity's approach of taking over all of your files. Scald also has far superior WYSIWYG integration.
The Asset module is another to consider. It's similar to Scald in terms of overall architecture but it hasn't seen as much maintenance so it's probably best to stick with Scald.
I'm not sure why you were down-voted. It is a good question to ask.
I would recommend the Drupal Media module. This one has the most integrations and allows you use views with exposed filters to be able to easily search through your media library. And easily catalog it with taxonomy. You can add content to the media items when you're selecting these from the browser, and even use the exif module to insert automated meta data upon upload.
Another option would be as Aaron mentioned, the Scald module But I have seen inconsistencies on how they handle different media items outside of WYSIWYG. Eg. Not being able to use it on a video file field.
In D8 there will be a module called Media Entity which integrates the different Media Browsers. That lists MediaBox and Asset too.