Can someone give me intuitive explanation of cofinality?
I pretty much understand what does it mean for a subset $x$ in $y$ to be cofinal (something like being dense when going from the left to the right), but when defining cofinality of $y$, one first says that
$$\text{cof}(y) = \text{min}\left\{\text{otp}(x) | x \text{ cofinal in } y\right\}$$
On the other hand, one states that this is the same as if we take $\text{card}(x)$ instead of $\text{otp}(x)$ in the above definition, but I wonder why..
The point is that a cofinality is always a cardinal. In other words, if $\alpha$ is an ordinal which is not a cardinal, then $\operatorname{cf}(\alpha)<\alpha$.
To see why, enumerate $\alpha$ by its cardinality, and start constructing a strictly increasing sequence which agrees on both the enumeration and the natural ordering of $\alpha$. You either get stuck at some ordinal below $|\alpha|$, or that you get that there is a cofinal sequence of length and order type $|\alpha|$.
If the cofinal sequence you ended up with was not of a cardinal order type, repeat the process. This is a decreasing sequence of ordinals, so it has to stop: exactly when the cofinal sequence is a cardinal, and it is not hard to check that this sequence is indeed cofinal in $\alpha$.
Note that it is often the case that "the smallest ..." ends up with a cardinal. It's not always, sure, but it isn't peculiar for cofinality. Other definitions include the Hartogs and Lindenbaum numbers.