What makes materials hard and strong? This is something I have wondered for a long time.  Why are some materials like steel, diamond, and even light materials like graphene stronger than others? Is it due to the strength of the electromagnetic/static fields that repel two binding atoms? Or is it something else?
 A: For crystal materials like diamond, this is a result of extremely stable bonds between the constituent atoms of the material. These bonds are only formed under immense pressure because inter-atomic repulsive forces keep the atoms from coming close enough to be forced to arrange themselves in that crystal structure.
Metals alloys like steel are much harder than elemental metals because they are a mixture of different elements like iron, nickel, chrome, etc. Basically, when a metal is composed uniformly of the same atom, they fit together very tightly and slide over each other easily. This is why pure gold is so soft and malleable.
A steel alloy, on the other hand, is composed of a few different elements. The different sized atoms mixed around in the bond structure cause irregularities in the atomic organization of the material, which makes it makes harder for slips and other deformations to occur.
A: If the material is hard to break, the bonds might be strong. If the material is light and strong, the material might have stronger bond, but mass density might be less, you can expect big voids in them. The materials hardness and strength depends on how bulk the material is, crystal structure and other factors.  
Spectrum Science, Deccan Herald, Tuesday, May 13, 2014.  

Graphene is the strongest, the thinnest material known to exist....It is not only the hardest material in the world, but also one of the most paliable. 
Only single atom thick, it has been called the wonder material....While the material was discovered a decade ago, it started to gain attention in 2010 when two physicists at the University of Manchester were awarded the Nobel prize for their experiments with it.      
The American Chemical Society said in 2012 that graphene was discovered to be 200 times stronger than steel and so thin that a single ounce of it could cover 28 football fields.
  ....In 2012, the American Chemical Society said that advancements in grephene were leading to touch-screen electronic that "could make cellphones as thin as a piece of paper and foldable enough to slip into a pocket.".....  

The paragraph points that a material can be strong being only single atom thick (not much bulky). 
A: If you are looking at elements and simple non-organic compounds (lets forget about plastics) you have two basic kinds of bonds. Ionic, like a salt crystal. The structure alternates sodium, chlorine, $\text{Na}$, $\text{Cl}$, $\text{Na}$, $\text{Cl},
\dots,+-+-+-$ in all directions. One is a little negative and the other is a little positive (Chlorine is a strong oxidizer - it steals electrons from the sodium). If you push on a salt crystal hard enough to displace a plane of atoms by $1/2$ the atomic lattice size - actually a little less - you cause $++$ and $--$ to line up and the electric field repulsion will make the crystal pop apart on a nice clean line. This is the electrostatic bond you were thinking about (good call).
The other bond popular among many atoms is the covalent bond. The idea is that electrons are shared to fill out orbitals with unpaired electrons or fill out complete orbitals. The atoms like this, and we can tell by how nice and stable they are - most of them anyway. The noble gases (argon neon etc) have full orbital shells and no need to play with others. The bond goes by other names like "molecular orbital bonds". This is the bond that explains why hydrogen comes in pairs $\text{H}_2$ and $\text{O}_2$ and $\text{H}_2\text{O}$ and $\text{CO}_2$, etc. Most compounds that are not crystals have covalent bonds. For an intuitive description this bond is more like a quantum mechanical bond than an electrostatic bond.
You asked specifically about steel, which is not a compound. Alloys are mixtures and quite complicated and some undergo fascinating changes when cycling temperature. As @Rick Sanchez says, alloys are a mix of stuff with different properties. Steel is like concrete with metal reinforcing bars. Different elements fill different spaces, like the gravel and sand and cement in concrete. The properties arise from how these parts all lock together and how much of each one there is.
