According to Stichtenoth's Algebraic Function Fields and Codes, the Hasse-Weil bound theorem is:
The number $N$ of places of $F/\mathbb{F}_q$ of degree one satisfies $|N-(q+1)|\leq 2gq^{1/2}$.
[here $q\in\mathbb{N}$ is a prime and $g$ is the genus of $F/\mathbb{F}_q$]
I've tried to check the theorem for some case when $g=0$, where we should have $N=q+1$. I took $y^2=x^2+1$ over $\mathbb{F}_3$, whose rational solutions are exactly $(0,1),(0,2)$, so $N=2\neq 4=q+1$.
What am I missing?
The missing points are in fact at infinity ! The curve you cosider is the restriction of the projective curve $$y^2=x^2+z^2$$ to the affine plane $z=1$. Be careful that the theorem applies to projective curves !
Hence the points $(1:1:0)$ and $(1:-1:0)$ are the missing ones !