What is Ideal and real sources?

An ideal voltage source imposes $u = E$ whatever the current; $E$ is its electromotive force (emf). An ideal current source imposes $i = I_N$ whatever the voltage. A real source (battery, generator, power supply) is modeled, in the generator convention, by

or, equivalently, by $i = I_N - u/r$ (Norton model: ideal current source $I_N = E/r$ in parallel with $r$); $r$ is the internal resistance. $E$ is the open-circuit voltage, $I_N = E/r$ the short-circuit current.