Quantum entanglement is one of the fundamental concepts in quantum theory, which states that a part of the system cannot be fully specified independently of the rest. Quantum entanlgement is a consequence of quantum superposition principle. For example, if two spin-$\frac 1 2$ particles are in a state: $\frac 1{\sqrt {2}}\Bigl (|\uparrow _1, \downarrow _2 \rangle + |\downarrow _2, \uparrow _1\rangle \Bigr )$ then the state of the first particle cannot be determined without the knowldege of the state of the second one. Thus the particles are entangled. This is in constrant to a state $|\uparrow _1, \downarrow _2 \rangle$ where the two particles are not entangled.
Quantum entanglement does not have an analogue in classical theory and is a basis of EPR and Schroedinger cat paradoxes.
Quantum entanlgement is essential for quantum computation and quantum communication. It can be quantitatevly characterized with various measures based on reduced density matrix of a subsystem.