Why is isobutane more thermodynamically stable than n-butane?
I would like to understand the term stability in organic chemistry.


We can speak of kinetic stability or thermodynamic stability when we discuss chemical compounds.  If a compound is kinetically stable, that indicates that it has a high barrier (large activation energy) to cross and is therefore less reactive than a compound that has a low barrier (small activation energy) to reaction.  In the following chart we can say that the reactants are kinetically less stable and the products are kinetically more stable since the barrier to the forward reaction is smaller than the barrier to the back reaction. 



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If one compound has a lower heat of formation than another compound, we can say that the former compound is thermodynamically more stable than the latter compound.  Again using the above chart, we can say that the product is thermodynamically more stable than the reactant.


  We know empirically that i-butane is more stable than n-butane. But
  why is it more stable?


Steric interactions do not provide the answer (reference).

We use arguments based on carbon-hydrogen hyperconjugation to explain many things.



For example, why more highly substituted double bonds are more stable than less substituted double bonds.  Carbon-carbon hyperconjugation can also be used to explain various chemical phenomena.  In the case at hand, let's compare carbon-carbon hyperconjugated resonance structures for n-butane and 2-methylpropane.



The hyperconjugated resonance structure for butane involves a primary carbocation, while the hyperconjugated resonance structure for 2-methylpropane involves a more stable secondary carbocation.  This more stable carbon-carbon hyperconjugated resonance structure explains the increased thermodynamic stability of 2-methylpropane compared to the isomeric n-butane.