What exactly was the state of armour at the close of the 13th century? As I understand, it would have been mostly mail, with perhaps greaves for the legs. But, were there any other parts of the body which were protected by plates? For that matter, where there other types of armours in use in western and central Europe besides mail at this stage?

A Coat-of-plates (or pair-of-plates) would have been worn over the hauberk, beneath the surcoat. Poleyns (Knee-cups) and schynbalds (simple greaves) protected the legs. Some kind of early plate gauntlets might also have been around

To add to what has been said:

-Great helms were fully developed
-Mail still in use for the whole body
-Prick spurs
-Poleyns of leather or metal
-Armoured surcoats or early coats of plates possible. Layers of aketon, mail, gambeson, surcoat popular, too
-demi-greaves/schynbalds of metal or leather
-Gauntlets are mentioned in the late 13th century, but most monuments/effigies show mail mittens
-Possibly gamboised cuisses to further protect the thighs

I do not refute the above replies regarding "state of the art" equipment at that time. What percentage of combatants were really equipped this way is an intriquing and probably an impossible question to answer.

A fair amount of art (versions of bibles, woodcuts, tapestries, etc.) depicted the battle of Courtrai (Golden Spurs) after it occurred in 1302. For the most part, it is difficult to tell much difference in the armour depicted in numerous of these examples of early 14th century art from illustrations of armour at the end of the 12th century.
http://www.liebaart.org/scenvr_e.htm All I am saying is that I suspect there were still lots of combatants equipped much as they were 100 years ago.

The chronology of effigies versus time in the link below is useful and in my opinion of what is really representative of the period. One must carefully differentiate between "end of the 13th century" and say the 1320's (now really reaching the end of the first quarter of the 14th century.)
http://www.themcs.org/armour/14th%20century%20armour.htm