structure of chitin
("Chitin" by Dschanz - own work (drawn with BKchem). Licensed under Public Domain via Wikimedia Commons)
If you look closely at the bonds linking the two 6-membered rings, you see that they are connected by a 1,4-beta linkage (where C#1 of one ring is connected to C#4 of the other ring). The groups attached to the N atoms are acetyl groups, so these are known as N-acetyl groups.
With lots of O and N atoms around the polymer, there are many opportunities for hydrogen bonding, which lend to the strength of chitin.
structure of hydroxyapatite Ca5(PO4)3OH
Electrostatic forces of attraction between the ions of hydroxyapatite contribute to its hardness and is commonly found in tooth enamel.
For a cool video on how the peacock shrimp smashes with its hammer claws and a more detailed explanation, I highly recommend Destin Sandlin at Smarter Everyday.
(By Jens Petersen (Own work) [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/) or CC BY 2.5 (http://creativecommons.org/licenses/by/2.5)], via Wikimedia Commons)
For more on the analysis and engineering of the hammer claw, here is a paper by James Weaver and co-authors: