Tags have long been available in OS X (well, certainly for somewhat longer than I've been a user) but the implementation was significantly limited. You could essentially assign a colour to a file or document, which would be a clear visual indication of whatever categorisation you had chosen to apply. This still exists in Mavericks, at least the basic principle, but it has been made far more useful.
One of the complaints about documents in iCloud has always been the lack of file structure access. The general idea is that individual apps will maintain their own document library, and you can always access that library through the app. In practice this isn't always useful; what if there are numerous documents for a project that are each managed by different apps? Well, tagging in Mavericks goes some way to providing a dynamic alternative to the hierarchical file system with some noted advantages and a lot of potential, and this is a brief overview of how I am starting to use it.
In the folder based file system a document exists within a specific folder and that's that. Tags present a view that behaves much like a folder, but rather than a file belonging to a folder any number of tags can belong to a document. Where you would have had to go to the one and only project folder before now you can access a document through any tag or combination of tags that have been applied to it. It's useful then to consider tags in some way similar to the levels of folders you might set up, each folder deeper narrows your focus to a smaller and more relevant group of files; each additional tag can do the same.
Tags come with a limited range of available colours (which seems like an odd restriction), they can also be colourless. I would recommend using colours only for those tags which are the broadest in scope or of constant relevance. Use these coloured tags in the sidebar on your finder. I have a coloured tag for my blog, and another for creative writing in general. I have a coloured tag for anything of specific personal relevance from letters to my résumé. It's then in colourless tags that things get more narrow. There are tags for individual writing projects, a tag for electronic receipts - things that would otherwise have had their own folder somewhere buried a level or two down. While only the most important, broadest or regular categories may be readily accessible at a click in the finder, the strength of tagging really shows in searches. You can search for tags using 'tag:keyword' in Spotlight and simply the keyword in Finder. More importantly you can search for a number of tags. So 'tag:project-name' combined with 'tag:infographics' then becomes a pretty useful way to getting to a very specific type of document and 'tag:writing' will let you see all your written documents from Pages, Byword, Textedit, Microsoft Word and so on all at once; even those stored in iCloud.
It may not yet be a file system replacement, but it has the potential to be far more convenient and efficient than the old nested folder approach to doing things. If you can find a method that works for you, something that will undoubtedly be more challenging for people with larger document volumes, then I don't think you'd be able to go back to just a folder based system.