Media in WordPress relates to pretty much anything you upload to the site, images and videos are the main ones, but it can include music or sound files, and documents (PDF, Word, Excel, etc.) and even zipped files.
The WordPress media library is where you can access all your uploaded media, and can see at a glance what posts they are attached to, date uploaded, who uploaded it and even if their are comments attached to it.
One thing I always miss is a quick way to access the files location, i.e. http://www.example.com/wp-content/random_images/logo.png, as sometimes its handy to be able to access the file direct.
In order to get this information, you click the Edit button.
If you click Edit on an image and then click Edit Image, you will find a basic cropping and resizing tool which is really useful.
There is no filter for finding your media, but there is a search system, if you know the file name or even just part of the name.
Obviously to add a new file. Since WordPress 3.3, there is the functionality that you can simply drag and drop a file here and it will upload, or you can click the button and manually find the file on your system.
You are not limited to uploading one file at a time, so if you know you will be using a lot of images for example, you can upload them all in one go (via drag and drop or select), and then attach them to the correct posts.