All the entanglement I've see is between 2 parties that have the same number of qubits. Is it possible to have entanglement where each party has a different number of qubits? Is it possible to have entanglement between 3 parties?


Sure! Look up multipartite entanglement.

An example of an entangled system among $n$ $d$-dimensional qudits is a GHZ state: $$|\mathrm{GHZ}\rangle\equiv\frac{1}{\sqrt d}\sum_{k=1}^d \lvert \underbrace{k,k,...,k}_n\rangle \equiv \frac{1}{\sqrt d}\sum_{k=1}^d \bigotimes_{j=1}^n \lvert k\rangle.$$ You can think of this as a generalisation of a Bell state, which you might have heard of.

Another notorious genuinely multipartite entangled state is a $W$ state. In the simplest case of three qubits this is $$|W\rangle=\frac{1}{\sqrt 3}(|001\rangle+|010\rangle+|100\rangle).$$

Entanglement between systems of different dimensions is also possible. For example, if you have a qubit and a qutrit, you can consider a state such as $$\frac{1}{\sqrt2}\left[|0\rangle\otimes\frac{1}{\sqrt3}\left(|0\rangle+|1\rangle+|2\rangle\right) + |1\rangle\otimes\frac{1}{\sqrt2}(|0\rangle-|1\rangle)\right].$$ A context in which you commonly encounter states with entanglement/correlation between a low-dimensional and a high-dimensional degree of freedom is for example a quantum walk (though one doesn't usually study the entanglement properties of the produced states when dealing with these).

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