There are several different ideas. A basic version is that entanglement of two particles (or attributes) means that their quantum state cannot be written as a product state. The most famous entangled states are the Bell states
https://en.wikipedia.org/wiki/Bell_state.
https://arxiv.org/abs/0910.4222v1
Quantum information: primitive notions and quantum correlations
Valerio Scarani
https://arxiv.org/abs/1604.01790
The Mathematics of Entanglement
Fernando G.S.L. Brandao, Matthias Christandl, Aram W. Harrow, Michael Walter