How Can I Know the Number of Points on an elliptic curve?
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.