Why is Kevlar stronger than nylon? They both contain Hydrogen bonding and should have equal strength but this is not the case instead Kevlar is fire resistant. What makes it stronger. Greater hydrogen bonds?!
Kevlar is an A,B co-polymer where the monomers are terephthalic acid and 1,4-diaminobenzene. The amide linkage, together with the aromatic rings in the polymer, makes a very rigid polymeric structure that stacks the chains in an organized way, maximizing the effectiveness of hydrogen bonding between amide linkages, together with pi-stacking of the rings: http://pubs.acs.org/cen/coverstory/8005/8005olympics.html
The chains form sheets that stack together along the axis of the fiber. In order for projectiles to break these fibers, these strong, inter-strand attractions must be overcome.
Standard Nylon is a similar polymer, where the aromatic moieties are replaced by aliphatic chains. These are much more flexible and give more degrees of freedom to the polymer chains. This results in a more random arrangement of chains that still hydrogen bond, but not in such an organized way, giving a weaker fiber.
First, it's important to define what you mean by "nylon", a term which is often used generically to refer to many different aliphatic polyamides. The most common are "nylon 6" and "nylon 6,6". Nylon 6 is the polymer of 6-aminohexanoic acid (aka 6-aminocaproic acid), while nylon 6,6 is from alternating units of 1,6-diaminohexane (aka hexamethylene diamine) and 1,6-hexanedioic acid (aka adipic acid).
Kevlar is one of the trade names of the polymer of alternating p-phenylenediamine and terephthalic acid. This is one example of the general class of aramid polymers, which means polyamides made from aromatic monomers.
As mentioned in the comments, both types of polyamide can form interstrand hydrogen bonds that strengthen the bulk material. However, the fact that kevlar is made entirely from rigid aromatic monomers means that the individual strands are much less flexible than the strands of the aliphatic polyamides. The aromatic groups can further participate in interstrand interactions that are stronger than the interstrand aliphatic interactions. Both of these factors contribute to making very strong and inflexible fibers. For this reason, Kevlar is used for ballistic protection applications.
On a side note, the fire resistance you mentioned is more closely associated with the aramid made from isophthalic acid and meta-phenylenediamine, which is marketed by Dupont under the name Nomex. The meta- orientation of the linkages means that the individual strands, while still rigid, are not linear, so they do not align in the same arrangement as in Kevlar. This leads to a much weaker material that is not effective for ballistics protection, but works well for making flame-resistant fabrics.