How Can I Know the Number of Points on an elliptic curve? If I have the following Elliptic Curve:
$$E: y^2 ≡ x^3 + 2x + 2   \mod 17$$

How Can I calculate the Number of Points on this elliptic curve $\#E$ ?

and how can I invest the following law which calculates the Number of Points on an Elliptic curve NPE: 

$$NPE=1+p+\sum_{x=1}^{p-1} \frac{f(x)}{p}$$
As the astute commenters pointed out, small cases can be done by hand, while Schoof's algorithm can be used for large fields.

By the way, there is the well known bound 
$$
q+1-2\sqrt{q}  \leq \#E \leq q+1+2\sqrt{q} 
$$
on what you call NPE, due to Hasse-Weil, for curves over $\mathbb{F}_q.$ If $q$ is prime, the inequality is strict.