A polyurethane binder is a resin built from urethane (carbamate) linkages formed by reacting polyols with isocyanates. In a printing ink or coating it performs three jobs at once: it disperses and holds the pigment, it forms a continuous film, and it bonds that film to the substrate.
Polyurethanes dominate solvent-based flexible packaging inks because the urethane backbone produces a film that is both tough and flexible — an unusual combination. Most polymers give you one or the other. That toughness-with-flexibility is exactly what a printed film needs to survive slitting, creasing, pouch forming and lamination without cracking or lifting.
The backbone chemistry matters. Polyester-based polyurethanes generally give higher film strength, better solvent resistance and better adhesion to polar substrates. Polyether-based polyurethanes generally give better hydrolytic stability and low-temperature flexibility. Which is correct depends entirely on the service conditions.