Explanation of the difference between the two orders
As shown in the second-tier header above, birds are generally subdivided into "passerines" and "non-passerines". What is the difference between the two? The answer is quite straight-forward, once one knows the Latin name of the common House Sparrow - "Passer domesticus". Passerines are birds that, like the House Sparrow, have three toes forward, one backward, e.g. when sitting on a perch, non-passerines don't.
Passerines
Brown Honeyeater as an example of a passarine (click on image for larger version)
Non-passerines
Australian Bustard as an example of a non-passarine (click on image for larger version)