Correlations are a property of a set of measurements. Two measurement outcomes are correlated if the associated probability distribution cannot be factorised, that is, when the outcome of one measurement gives information about the outcome of the other measurement.
The concept of "correlations", in this sense, is not inherently quantum, although quantum mechanics can make for correlations stronger than those allowed by classical probability theory.
On the other hand, entanglement is a property of a state, with respect to some partition on the underlying space. A bipartite state $\rho$ is said to be entangled if it cannot be written as a convex combination of product states, that is, if it cannot be written in the form $\rho=\sum_k p_k \rho_k^A\otimes\rho_k^B$ for some $p_k\ge0, \sum_k p_k=1$ and states $\rho_k^A,\rho_k^B$.
The bipartite structure is usually, although not necessarily, taken to refer to degrees of freedom of spatially separated particles. It can however refer to any pair of degrees of freedom of a quantum system.
Entangled states can produce nonclassical correlations, but this is not necessarily the case. For example, not all entangled states can produce Bell violations.
On the other hand, entangled states always display some form of correlation: given a pure entangled state $|\psi\rangle$, write it in its Schmidt decomposition as $|\psi\rangle=\sum_k \sqrt{p_k} |u_k\rangle\otimes|v_k\rangle$. Then, measuring in the $\{|u_k\rangle\}_k$ basis on the first space and in the $\{|v_k\rangle\}_k$ basis in the second will give correlated outcomes (the $k$-th outcome for the first party implies that the second party also will have measured its $k$-th outcome).
