Hydrogen cloud at the universe's beginning? What prevented all of the hydrogen at the universe's start from coalescing into one gigantic star?
 A: There are two reasons:
First, the expansion of space, which was rapid in the early universe, separated the initial density fluctuations into isolated potential wells. Dark matter and ordinary matter then accumulated into these local potential wells, which eventually become galaxy clusters.
Second, the temperature of the early universe was very high, so that hydrogen and helium atoms were ionized. These ionized atoms and free electrons interacted with photons; basically, the early universe was like a plasma. But these photons exerted a radiation pressure on the matter: as matter falls into the potential wells, it heats up, and the radiation prevents it from compressing further. In fact, the radiation reverses the motion, and the matter recoils. 
As the universe expanded, the photons lost energy, and matter could fall back in, etc. This 'tug of war' between matter and radiation caused oscillations, and continued until the universe had cooled sufficiently for the ions and electrons to combine into neutral atoms. The photons then decoupled from the atoms, and we can still observe them today as the Cosmic Microwave Background. The oscillations are imprinted in this CMB as temperature fluctuations.
For a detailed overview of the processes in the early universe, see this source: http://background.uchicago.edu/~whu/intermediate/intermediate.html
A: Pulsar and Chris White have given nice explanations of why the dynamics of the early universe would not lead to the formation of one big starlike object. There is also another, much more generic argument against such a process, which boils down to the existence of cosmological horizons. At any given time in the evolution of the universe, there have been parts of the universe far enough apart that they could never have had any causal relationship -- no signal could have propagated between them, even at the speed of light, even if the signal was emitted immediately after the big bang. If matter in region A and matter in region B are separated in this way, then it's not possible for them to have collapsed into the same object, simply because relativity prevents the matter at A and B from even coming together that fast.
