Both of these terms depend on what basis you're in, which can make them a little arbitrary. For example, I think of a single particle in the state $|\uparrow\rangle + |\downarrow\rangle$ as in a superposition, though someone else who prefers the $\hat x$ basis may disagree and call it their eigenstate (we're both right). Similarly, entanglement is something we only understand if we work in the single-particle basis. There, we define an entangled state as a state that cannot be written as a product of single-particle states. By a single-particle state, I mean something that can be written as $|\psi\rangle_1 \otimes |\psi \rangle_2$ for two particles.
For example, $|\uparrow\rangle_1 |\uparrow\rangle_2$ is not entangled, since it's just the product of $|\uparrow\rangle_1$ and $|\uparrow\rangle_2$. This is a very correlated state, but it's not entangled. Here, I'm using the subscript to denote wihch particle. An example of a state that isn't entangled is
$$ |\uparrow\rangle_1 |\uparrow\rangle_2 + |\uparrow\rangle_1 |\downarrow\rangle_2 + |\downarrow\rangle_1 |\uparrow\rangle_2 + |\downarrow\rangle_1 |\downarrow\rangle_2$$
this is because I can write it as
$$(|\uparrow\rangle_1 + |\downarrow\rangle_1) \otimes (|\uparrow\rangle_2 + |\downarrow\rangle_2)$$
On the other hand, the following states are entangled:
$$ |\uparrow\rangle_1 |\uparrow\rangle_2 + |\uparrow\rangle_1 |\downarrow\rangle_2 + |\downarrow\rangle_1 |\uparrow\rangle_2 - |\downarrow\rangle_1 |\downarrow\rangle_2$$
$$ |\uparrow\rangle_1 |\uparrow\rangle_2 + |\downarrow\rangle_1 |\downarrow\rangle_2$$
The latter is known as a Bell state. You cannot write either of these states as a generic $|\psi\rangle_1 \otimes |\psi \rangle_2$: go ahead and try!
A more practical metric I use is: does the result of measuring the state of one particle change my expectation of the state of another? If yes, the particles must be entangled. (Warning: if the answer is no, it doesn't rule out entanglement! You might need a more clever experiment). For example, if I start with $|\uparrow\rangle_1 |\downarrow\rangle_2$, and I know I will measure that the second particle is $\downarrow$, regardless of what basis I measure the first particle in. However, if I start with the state $|\uparrow\rangle_1 |\uparrow\rangle_2 + |\downarrow\rangle_1 |\downarrow\rangle_2$, I have $50-50$ odds measuring that the second particle is $\uparrow$ or $\downarrow$. If I measure that the first particle is $\uparrow$, suddenly I know that I'll measure the second particle is $\uparrow$. I've learned something about the second particle only by measuring the first. A measurement on the first particle has changed the odds on the outcome of measuring the second, the hallmark of entanglement.
Side note: if a friend places two marbles in a bag, and promises that both are red or both are blue, you'd get a similar result as my second experiment. But this is not entanglement, that's just a classical lack of knowledge! My proposed experiment to measure entanglement of $|\uparrow\rangle_1 |\uparrow\rangle_2 + |\downarrow\rangle_1 |\downarrow\rangle_2$, by measuring each particle in the $\hat z$ basis is convincing only if you know that you started in a pure, quantum mechanical state. In reality you need a more clever experiment to show that you have an entangled state. Examples include Bell's inequality and the CHSH inequality.