What makes carbon special and versatile?
There are two key factors that account for the ubiquity of carbon compounds.


Bond strengths: Look at the following table of bond strengths and notice how the strength of both carbon-carbon single and double bonds is much greater than the bond strengths found in other molecules.


\begin{array}\hline
Typical~Single ~Bond~ Dissociation ~Energies~ (kcal/mole)\\ \hline
\end{array}
\begin{array}{|c|c|c|}\hline
C-C &amp; 83    \\ \hline
N-N &amp; 38    \\ \hline
O-O &amp; 35    \\ \hline
Si-Si &amp; 52    \\ \hline
P-P &amp; 50   \\ \hline
S-S &amp; 54    \\ \hline
\end{array}

\begin{array}\hline
Typical~Double ~Bond~ Dissociation ~Energies~ (kcal/mole)\\ \hline
\end{array}
\begin{array}{|c|c|c|}\hline
C=C &amp; 146    \\ \hline
N=N &amp; 109    \\ \hline
O=O &amp; 119    \\ \hline
P=P &amp; 84   \\ \hline
\end{array}
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These numbers tell us that carbon-carbon bonds are significantly more stable than other bonds and that the driving force for their formation will be more exothermic.  Therefore, large collections of carbon atoms can be bonded together to produce very stable systems.


Hybridization: Saying that carbon is tetravalent doesn't fully convey the versatility carbon can display in forming bonds. Hybridization involves a re-mixing of s- and p-orbitals in response to bond formation and this leads to the formation of more stable bonds. Carbon is capable of extensive variation in the hybridization of the bonds it forms, much more so than most other atoms.  This wide range in hybridization that is possible for carbon allows it to form much more stable bonds in a wide variety of different structural situations.
\begin{array}\hline
Typical~Carbon-Carbon ~Bond~ Strengths~ (kcal/mole)\\ \hline
\end{array}
\begin{array}{|c|c|c|}\hline
alkane &amp; sp^3 &amp; 84    \\ \hline
alkene &amp; sp^2 &amp; 150    \\ \hline
alkyne &amp; sp &amp; 200    \\ \hline
\end{array}