Can matter be created from energy? 
The small, hot, dense early universe the size of an atom was made up entirely of energy, it wasn't until after the expansion began and the universe cooled down some of that energy began converting into the first atomic nuclei.

This quote seems a little dubious but I think this is worth asking anyway:
Can someone explain the atomic process, if it even exists, of how this would work to convert energy in to matter, and what form of energy was initially present, and what is required to cause this change?
 A: Can someone explain the atomic process, if it even exists, of how this would work to convert energy in to matter, and what form of energy was initially present, and what is required to cause this change?
It is not an atomic process, it is an elementary particle process, atoms are made up of elementary particles in a non trivial way.
At the level of elementary particles we are talking of special relativity and mass and energy are interchangeable depending on the process under consideration.
In the macro dimensions we live in one may consider light as carrying energy without being massive, but at the elementary particle level the light beam is made up of a huge number of photons, particles with zero mass.
A photon photon scatter can create particles, as shown in this link Even a gamma gamma collider is discussed to study these higher order interactions in creating particles.
Now at the very early times of the Big Bang there are models, essentially extrapolated expecting similar diagrams to hold in the very small dimensions, of elementary particles scattering off elementary particles and generating more of the same, possibly to start with very exotic particles not (yet?) known in our labs. They will be a soup of energy to matter and matter to energy. 
The Big Bang theory postulates that as the system expands the universe cools and the energy levels evolve in this soup to become  a "photon quark gluon" soup and with the further expansion finally baryons and electrons appear and photons can escape binding in the soup and give us currently the cosmic microwave background snap shot.
A: They are actually trying to do so in the lab. They need a very potent laser (I believe they didn't reach the critical power yet).
The idea is simple, inside the lase cavity you generate a very powerful electromagnetic field, powerful enough such that the photos have enough energy to transform a virtual electron-positron pair into a real one.  See this link for more detail:
http://www.kurzweilai.net/scientists-discover-how-to-turn-light-into-matter-after-80-year-quest
