Any state which cannot be decomposed into an exact tensor product of states over the state spaces of the subsystems is an entangled state. For example, any 2-qubit state which cannot be decomposed into $|x\rangle|y\rangle$ form is an entangled state.

Any state which cannot be decomposed into an exact tensor product of states over the state spaces of the subsystems is an entangled state. For example, any 2-qubit state which cannot be decomposed into $|x\rangle|y\rangle$ form is an entangled state.

As for the 2nd part of the original question:
Any local operation such as applying a Pauli-X gate can only preserve or destroy the entanglement of the original pair of qubits. Local operations (LOCC) cannot be used to create entanglement.
As an example one may consider the Bell states and observe the effect of applying the Pauli operators on any of the individual qubits in the entangled pair.