Are $\ell^p$ spaces a special case of $L^p$ spaces? My professor said that $\ell^p$ spaces are $L^p$ spaces with a discrete measure. But how can that be true if that the inclusion of the spaces is in different directions in the two cases? In the first case, $\ell^p \subset \ell^q$ if $p < q$, while $L^q \subset L^q$ if $p < q$!

The proof of the inclusion $L^q(\mu) \subset L^p(\mu)$ for $p<q$ you saw probably assumed that $\mu$ is a finite measure.  This inclusion does not hold in general (consider $\mu$ the Lebesgue measure on $\mathbb{R}$).
What your professor was hinting towards was the fact that $\ell^p=L^p(\mu)$, with $\mu$ the measure on $\mathbb{N}$ (with $\sigma$-algebra the power set of $\mathbb{N}$, hence the term discrete measure) given by
$$\mu(A)=|A|,$$
where $|A|$ denotes the number of elements of $A$.
