Lithium and magnesium are Group 1 and Group 2 elements respectively. Elements of these groups are highly ionic, and I've never heard of them forming significantly covalent inorganic compounds.
Yet these elements form a variety of organometallic compounds ($\ce{PhLi}$, the whole family of Grignard reagents, etc). Organometallic compounds have significant covalent character (i.e., the bond can be called covalent) in the carbon–metal bond.
What's so special about carbon that makes these elements form covalent bonds?
Answer
The character of the bond is determined by the electronegativity of the atoms.
Speaking of bonds as purely ionic or covalent is not always correct - usually it is more correct to say that a bond has ionic or covalent characteristics.
So comparing the difference in electronegativities gives us the following:
$$\begin{array}{cc}\hline \text{Difference in electronegativity} & \text{Type of bond} \\ \hline < 0.4 & \text{Non-polar covalent} \\ 0.4 \mathrm{-} 1.7 & \text{Polar covalent} \\ >1.7 & \text{Ionic} \\ \hline \end{array}$$
At the upper end of the polar covalent spectrum, the bonds frequently have both covalent and ionic characteristics.
For organometallic compounds, the difference in electronegativity between Li and C is $1.57$. So while this is still in the polar covalent range, it is also close to ionic. Similarly, the difference between Mg and C is $1.24$ - again, a very polar covalent bond.
Compare this to the difference between H and C ($0.35$) - a non-polar covalent bond.
So to answer your question, the thing that is "special" about carbon is that it has a fairly low electronegativity compared to the chalcogens and halogens. Granted, bonds with carbon are also going to be weaker than in say LiCl, but that's what makes organometallic compounds actually work to form carbon-carbon bonds.
(The electronegativity values came from wikipedia's great chart)
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