Let $X,Y,Z$ be random variables and $p(X)$ be pdf of $X$ and $p(Y|X)$ be conditional pdf of $Y$ given $X$.
I have no problem dealing with $p(X)p(Y|X) = p(X,Y)$: joint distribution and $\int_x p(X)p(Y|X) dx= p(Y)$.
However, as one more variable is conditioned, I have no idea how to handle this and the example I have is $p(X)p(Y|X,Z)$ and $\int_x p(X)p(Y|X,Z) dx$.
Could someone explain what those mean?
You can generalize $p(x) p(y \mid x) = p(x,y)$ to $$p(x \mid z) p(y \mid x, z) = p(x,y \mid z)$$ by conditioning each term on $Z$. This leads to $$\int p(x \mid z) p(y \mid x,z) \, dx = p(y \mid z).$$
In general I don't think your expression $\int p(x) p(y \mid x,z) \, dx$ can be simplified. However, it may be that in your context, you have some assumption that $x$ is independent of $z$, in which case $p(x) = p(x \mid z)$.