TITLE: Weil's Riemann Hypothesis for dummies?
QUESTION [12 upvotes]: Weil's Riemann Hypothesis is a deep result that I don't fully understand, but it has understandable corollaries which interest me. For example:
(a) For any projective curve $X$ satisfying certain conditions, the number $N$ of points in $X$ with coordinates in $\mathrm{GF}(q)$ satisfies $|N-(q+1)|\leq\mathrm{const}\cdot\sqrt{q}$. (The deviation is $0$ when $X$ is a projective line.)
(b) For any nontrivial multiplicative character $\chi$ on $\mathrm{GF}(q)$ and any polynomial $f$ of degree $n$ satisfying certain conditions, we have 
$$\bigg|\sum_{x\in\mathrm{GF}(q)}\chi(f(x))\bigg|\leq(n-1)\sqrt{q}.$$
Questions:

Is there a reference (legible to an English-speaking non-expert in the field) which gives the rigorous statements of these corollaries? In particular, I would like conditions which one can verify without a background in algebraic geometry.
Are there other corollaries of Weil's Riemann Hypothesis which are also widely understandable? EDIT: I'm mostly interested in the Riemann Hypothesis, but I'm also happy to learn understandable consequences of the other Weil conjectures and related results.

REPLY [9 votes]: This is an explicit version of part (a) of Gerry Myerson's answer.  If $f(x,y)$ is an absolutely irreducible polynomial in $\mathbf{F}_q[x,y]$ of total degree $d>0$, and $N$ is the number of zeroes of $f(x,y)$ in $\mathbf{F}_q\times\mathbf{F}_q$, then
$$
q+1-(d-1)(d-2)\sqrt{q}-d\le N\le q+1+(d-1)(d-2)\sqrt{q}.
$$
Likewise, if $f(x,y,z)$ is an absolutely irreducible homogeneous polynomial in $\mathbf{F}_q[x,y,z]$ of total degree $d>0$, and $N$ is the number of zeroes of $f(x,y,z)$ in $\mathbf{P}^2(\mathbf{F}_q)$, then
$$
\lvert N-(q+1)\rvert\le (d-1)(d-2)\sqrt{q}.
$$
These results comprise Corollary 2 in the paper "The number of points on a singular curve over a finite field" by Leep and Yeomans (Arch. Math. 63 (1994), 420-426).