Let's speak in the classical context (non quantum). We assume that point charges move in a conductor following Newtonian mechanics. How do point charges move along the boundary of the conductor and how do they stop (equilibrium) in the end?
There is a potential barrier at the boundary (surface) of a conductor, knowing as the work function. Work function is defined as the energy requied to remove an electron from inside the metal to the vacuum.
The typical number of work function for metal is about $4 eV$ ($3-5 eV$) which forms a barrier for electron to escaping the metal (due to effects of electron-ion attraction and the electron-electron exchange). The surface reigion is about $10 \dot A$, forming an electric field of intensity $10^7 volts/cm$ at the boundary of a metal to prevent the electron from escaping.