The following is a quote from Wolfram MathWorld article about prime ideals.
A maximal ideal is always a prime ideal, but some prime ideals are not maximal. In the integers, $\{0\}$ is a prime ideal, as it is in any integral domain. Note that this is the exception to the statement that all prime ideals in the integers are generated by prime numbers. While this might seem silly to allow this case, in some rings the structure of the prime ideals, the Zariski topology, is more interesting. For instance, in polynomials in two variables with complex coefficients $\mathbb C[x,y]$, the ideals $$\langle 0\rangle \subset \langle y-x-1\rangle \subset \langle x-2,y-3\rangle$$ are all prime.
How can we prove that $\langle y-x-1\rangle$ and $\langle x-2,y-3\rangle$ are prime ideals in $\mathbb C[x,y]$?
Perhaps showing that quotient rings are $\mathbb C[x,y]/\langle y-x-1\rangle\cong \mathbb C[x]$ and $\mathbb C[x,y]/\langle x-2,y-3\rangle\cong\mathbb C$ would be way to go? (This would mean the quotient ring is an integral domain and therefore the ideal is prime.)