In a piece of metal, what holds the atoms together?

In a metal like sodium, for example, each #"Na"# atom is touching eight other #"Na"# atoms.

Each atom shares its #"3s"# orbital with its eight neighbours.

You end up with a giant set of molecular orbitals extending over all the atoms.

The result is that the valence electrons are free to move throughout the metal.

The metal atoms that lose their electrons become positive ions, and they are embedded in a "sea" of electrons that is free to move throughout the solid.

The ions are attracted to the sea of electrons around them, and the electrons are also attracted to them.

These attractive forces hold the metal together in one piece.

We call them metallic bonds.

The standard description of #"metallic bonding"# is #"positive ions in a sea of electrons"#. In an array of close-packed metal atoms, each metal atom donates one or two or more electrons to the metallic lattice. The electrons thus form the glue that extends over the entire metallic lattice.

Because the electrons are non-localized, and the structure is non-molecular, metals tend to be (i) highly #"malleable"#, i.e. capable of being beaten into a sheet, and (ii) highly #"ductile"#, i.e. capable of being drawn out into a wire. Also (iii), metals tend to be electrically and thermally conductive due to the delocalized electrons.

These first 2 properties, which are conferred by metallic bonding, make metals the premier material with which to build tools.

I was to going to include a picture, but I see Ernest has already posted an excellent picture. Apologies for cross posting.