After the collapse of a nebular cloud, atoms begin gravitating together to form a condensed center. What happens next in the star formation process?

The condensed center will act as a central mass that will supply sufficient gravitational pull to continue to gather atoms around it.  As the mass becomes larger, the atoms at the core (likely hydrogen as this is a nebula) will be subject to greater and greater pressure.  What happens next depends on the amount of mass that has been gathered.

Scenario 1: Nuclear fusion.  This is where a star is first "born."  All stars go through the process of fusion--combining smaller atoms into larger.  In our sun, as an example, two hydrogen atoms are combined into helium.  This is seen as the primary nuclear reaction and it gives off a tremendous amount of energy.  We benefit from this in the form of light and heat.  Secondary (and further) reactions will occur later in the star's life, leading to progressively heavier elements, like iron.  The ultimate fate of a star (like neutron stars, supernovae, black holes) depends on the interaction and amount of these heavier elements.

Scenario 2: No nuclear fusion.  This is how you get gas giants like Jupiter.  Although Jupiter has a generous amount of matter, the forces at the core will not be sufficient to ignite fusion.  Planets will continue to gobble up planetary material (just look at how Jupiter impacted planetary formation in our solar system), but will never become luminous.

Scenario 2 (b), the coolest one: Close to, but no nuclear fusion.  This is what is known as a brown dwarf.  These failed stars have twice the mass of Jupiter (bear in mind that Jupiter has more mass than all of the other planets in our solar system combined), but less than the nuclear threshold, thought to be 0.08 times the mass of our Sun.  Though these will never go through fusion and hence be labeled a star, they have masses too large to be classified planets.