metal-carbon bond
Imagine the periodic table as a city with two old neighbourhoods that rarely talk to each other: the inorganic side, home to metals and salts, and the organic side, home to carbon and everything built from it. A metal-carbon bond is a bridge thrown directly across the river between them. The moment a carbon atom is joined straight to a metal atom, you have left both old neighbourhoods and arrived in a third district called organometallic chemistry, where the rules of both sides meet and a great deal of modern catalysis happens.
More precisely, a compound is called organometallic when it contains at least one direct bond between a carbon atom and a metal (or metalloid) atom. The carbon may belong to a simple alkyl group such as a methyl, to an aromatic ring, to a single carbon monoxide molecule, or to a flat ring like cyclopentadienyl that lies face-on to the metal. The character of the bond runs across a spectrum. With very electropositive metals such as sodium or magnesium the bond is highly polar, almost ionic, leaving carbon with a large share of negative charge. With the softer transition metals such as iron, rhodium or palladium the bond is far more covalent and well-behaved, which is exactly why those metals are the workhorses of catalysis. Note that bonds from a metal to oxygen or nitrogen, as in a carbonate or an amine complex, do not count: the carbon must be bonded to the metal itself.
This bridge matters because it is where chemists learned to make a metal grab an organic fragment, rearrange it, and hand it back transformed. Almost every industrial homogeneous catalyst, from the rhodium that turns alkenes into aldehydes to the palladium that stitches carbon skeletons together, lives or dies on the making and breaking of metal-carbon bonds. It is worth remembering that inorganic chemistry was never lifeless or carbon-free; carbon shows up in carbonates, carbides, carbon monoxide and these organometallics, and the metal-carbon bond is one of the most productive places where the inorganic and organic worlds overlap.
Tetramethyltin, Sn(CH3)4, has four covalent tin-carbon bonds and is a stable, distillable liquid. Methyllithium, CH3Li, also has a metal-carbon bond, but lithium is so electropositive that the bond is strongly polarized toward carbon, making the methyl group behave almost like a free carbanion that snatches protons greedily.
The same kind of bond, M-C, ranges from nearly covalent to nearly ionic depending on the metal.
A bond to carbon is what defines an organometallic; a metal bonded only to oxygen or nitrogen donors, even of an organic molecule, makes a coordination compound, not an organometallic one.