Why are free electrons free? This is what I understand so far: in a conductor, the ions have a weak pull on the valence electrons. So when an electric field is applied, the free electrons are able to easily move about. Makes sense.
In a neutral conductor with no electric field, the free electrons aren't bound to any ions. Why? I understand that the ions have a weak pull on the electrons, but what makes electrons leave the ion and stay free?
 A: An atom in isolation offers a potential well, and electrons form bound states in the well. The energy of those bound states can be calculated exactly in the case of a single-electron (hydrogen-like) atoms or by variational computational methods for more complicated cases.
Now when you put several atoms together in a tight and regular array, they offer a combined well resulting from the sum of all their potentials.
That combined well might look (in cartoon form) something like this:

Here the red lines represent electron energy level that are still contained by the locally stronger effect of their "own" nucleus, but the blue line represents those slightly higher energy levels that see the combined potential as a single large energy well (with some high-frequency detailed structure).
In a conductor, non-conduction electrons fill all the red (local) energy levels and the remaining electrons must (because of Pauli exclusion) then occupy the blue (non-localized) conduction levels.
A: In a single free atom, electrons have well defined energy levels and are somewhat bound to atom. Consider the following quantum mechanical model of atom to get an idea about an isolated atom.

When all this isolated atoms come together to form the crystal, the atoms do not have well defined energy levels.  There will be molecular orbitals. When the atoms get even closer, energy bands are formed which are continuous variation of energy levels. Continuous energy level mean electrons are somewhat free to move. Energy bands splits into valence and conduction band. These two bands are separated by certain energy gap. Depending on this energy gap, materials are distinguished as conductors, insulators, and semiconductors.   
The energy required to attain energy level of conduction band is obtained by applying even small electric field in conductors. So, formation of continuous variation of energy levels makes electrons to be free. 
A: The intuition is that the valence electrons are so far away from their nucleus that when they combine to form metals, they feel the attraction of all the other nuclei as strongly as from theirs.
In a more rigorous description, the orbitals for the valence electrons fully overlap with their neighbouring atoms, so their "play field" extends all over the material. 
