At the beginning, before the Big Bang, all the matter and energy in the universe was located in a single, densely-packed space. At the moment of the Big Bang all of that matter and energy began to expand. The universe at the time was extremely hot (estimates are about 1012 degrees Kelvin). Roughly 30 minutes after the Big Bang, Helium and Hydrogen were formed, likely because they are the simplest elements. For a very long time, these elements were the only ones to exist in the entire universe.

As the universe grew older, it began to cool down. When the universe was about 700,000 years old it was cool enough for electrons to attach themselves to protons and neutrons so heavier elements could form. To better understand why, think of the universe like a giant pot of soup. After the Big Bang, the universe continued to "simmer." That means particles were moving about rapidly, and constantly coming into contact with each other. With that contact, the elements such as Deuterium (which is like hydrogen) and Lithium formed.

Sources

http://www.sci.uidaho.edu/geol423/pdf/lectures_pdf/Topic_1_423_2009_bw%20[Compatibility%20Mode].pdf (fair use)
http://www.astro.utoronto.ca/~bolton/sci199y/h0_web/img13.html (fair use)
http://spiff.rit.edu/richmond/asras/chemcomp_iii/chemcomp_iii.html (fair use)