why we call it unnormalized posterior density

Bayes' rule yields the posterior density:

$$p(\theta| y) = \frac{p(\theta| y) }{p(y)} = \frac{p(\theta)p(y| \theta)}{p(y) }$$

Since the factor $p(y)$ doesn't depend on $\theta$, then we can drop it and write it:

$$p(\theta | y) \propto p(\theta)p(y| \theta)$$

We call the right side as unnormalized posterior density. Can anyone tell me why it is called unnormalized?

The posterior density must satisfy $ \int p(\theta \mid y) \, d\theta = 1$.

If you work instead with $p(\theta) p(y \mid \theta)$ instead, this function of $\theta$ is the same as $p(\theta \mid y)$ except for the normalizing constant $1/p(y)$. In order to turn $p(\theta) p(y \mid \theta)$ into a density you must multiply by this normalizing constant $\frac{1}{\int p(\theta) p(y \mid \theta) \, d\theta}$ which may not equal $1$.

In situations where you only care about ratios like $\frac{p(\theta \mid y)}{p(\theta' \mid y)} = \frac{p(\theta) p(y \mid \theta)}{p(\theta') p(y \mid \theta')}$ you don't need to compute this normalizing constant, which is why the unnormalized density can be useful, especially when the integral $p(y) = \int p(\theta) p(y \mid \theta) \, d\theta$ is hard to compute.