Basically the deal with Kontakt Player is it can only play protected libraries. If a company makes a sample library, they can choose to protect it with NI's activation/DRM. To do this, they have to pay a license fee to NI for each copy sold. However it gets their library protected (not that it is good protection, but there you go) and it also makes it able to work in Kontakt Player. Another way to look at it is they are licensing a copy of Kontakt Player.
As such, any libraries thus protected work with it, and you can download it and use it for them freely.
Not all libraries you buy are like that though. Some companies protect all their libraries, some companies none, some a mixture. It can depends on a number of things. Some companies don't do protection because they don't want to pay the license fee. Sometimes it is because more complex libraries can't work in the limits of the Player (since it doesn't have all the effects and so on). Sometimes it is because professionals like the ability to edit libraries, which is diminished in protected libraries.
Regardless, it is generally very well noted. If a library needs the full Kontakt, it'll say so, if Player is ok, it'll say so and usually advertise that it is included (which doesn't matter as you can download it separate).
So for example Soundiron's
Olympus Elements, their lighter weight choir, works with Kontakt Player. However
Mars their full featured men's choir needs the full Kontakt to work.
If you own a Kontakt Player library, you can upgrade to Kontakt at a reduced price of $250. The full version of Kontakt works with all the libraries that work in Player, as well as ones that require the full version.
Also: Old libraries that use Kompakt work with Kontakt Player. Basically Kompakt was the predecessor to Player. Back in the day they had two different products. They decided to unify the codebase, but just have different feature sets. So if you own a Kompakt library, you can download Kontakt Player and it'll work happily.
Komplete is then a super-set of Kontakt. It includes the full version of Kontakt, as well as the full version of a number of other NI programs (like Guitar Rig, Battery, Reaktor, FM8, Massive and so on) and also a bunch of NI's sample libraries for Kontakt. If you want Komplete, it is generally best to buy it, rather than buying Kontakt and upgrading. Komplete is normally $500, and the crossgrade from Kontakt to complete is normally $400 so you don't really get a discount for owning Kontakt already.