I got into a discussion about energy in various fuels.  In this
discussion, I stated that I believe the hydrogen is the primary energy
source in various fuels such as oil, coal, natural gas, and other
fuels.

My friend felt that the carbon was the primary energy source in coal.  

Please help me answer this discussion.

EEnergy is stored in fossil fuels (or any other type of fuel) in the
form of the bond energy between the atoms making up the molecules in a
particular fuel.  It is therefore not meaningful to ask whether a
specific element ("hydrogen" or "carbon") is the source of energy in a
fuel.  Instead, you should be asking "which bonds between which two
types of atoms" are the source of the energy.

The net thermal energy released by combustion of the fuel is given by
the difference between the sum of the bond energies of the molecules
in the combustion products and the sum of the bond energies of the
molecules in the fuel.

With regard to your specific question, different fossil fuels have
radically different chemical compositions and molecular makeup. 
Natural gas is primarily methane, or CH4 (with some additional "light"
hydrocarbons, see <http://www.uniongas.com/aboutus/aboutng/composition.asp>).
 In the limit of pure methane, *all* the energy released by its
combustion comes from the breaking of carbon-hydrogen bonds, because
that's the only type of bond present in this molecule.  C-H bonds have
a bond energy of about 410 kJ/mole
(<http://www.wou.edu/las/physci/GS361/Energy_From_Fossil_Fuels.htm>).

Petroleum is a complex mixture of longer-chain and cyclic
hydrocarbons.  Some of the energy comes from breaking C-H bonds, and
the rest from breaking C-C bonds of various types (single, double,
triple, and aromatic bonds).  As a rough guide, one can take the
carbon/hydrogen ratio of petroleum to be about 2.  In the case of
something like gasoline, which has the approximate composition of
octane (C8H18, with a C/H ratio of 2.25), there are 7 C-C bonds, and
18 C-H bonds.  As in the case of methane, each C-H bond contributes
about 410 kJ/mole and each C-C bond contributes about 350 kJ/mole, so
the energy stored in the C-H bonds amounts to 7380 kJ/mole of octane,
while the energy stored in the C-C bonds amounts to 2450kJ/mole
octane. Therefore, about 75% of the energy content of octane is in the
C-H bonds, and only 25% in the C-C bonds.

As the carbon/hydrogen ratio of the fuel continues to increase, the
fraction of the total energy present in the C-C bonds (relative to C-H
bonds) also increases, but because the bond energy of the C-C bond is
less than that of a C-H bond, the energy density of the fuel (kJ/gram
fuel) decreases.  Coal, with a C/H ratio of about 1/1 has the lowest
energy density of the common fossil fuels.

Thank your, hfshaw-ga for an excellent comment.  As far as I'm
concerned, you answered my question and pointed out the fact that the
energy is stored in the form of bonds.