3-SAT and Systems of Nonlinear Modular Equations
The answer depends on whether we're talking about a single equation, or a system of equations; and whether the modulus $m$ is prime or composite.

A single equation

When the modulus $m$ is prime, then you can find a solution to a single equation

$$a_n x^n + \dots + a_1 x + a_0 = 0 \pmod m$$

in polynomial time (if one exists) by factoring the polynomial $a_n x^n + \dots + a_0$ over the finite field $\mathbb{F}_m$.  When $m$ is composite, finding a solution is (in general) at least as hard as factoring $m$.  See e.g.,Is deciding if there&#39;s a solution to a single multivariate quadratic equation NP-hard?.

System of equations

That's the situation for a single equation.  In contrast, finding a solution to a system of (multiple of) these equations (in multiple unknowns) is NP-hard, regardless of whether $m$ is prime or composite.  There's a simple reduction, if $m$ is prime.  Each clause is translated to a nonlinear equation, e.g., $(x_i \lor x_j \lor x_k)$ translates to the equation

$$(1-x_i)(1-x_j)(1-x_k)=0 \pmod m.$$

As another example, $(\overline{x_i} \lor x_j \lor x_k)$ translates to

$$x_i (1-x_j)(1-x_k)=0 \pmod m.$$

Finally, for each variable $x_i$ we add the equation

$$x_i (1-x_i) = 0 \pmod m$$

to force $x_i$ to be 0 or 1 (this is unnecessary when $m=2$).

An extension of this idea works when $m$ is composite, too; if $m=pr$ where $p$ is prime, then multiply the left-hand-side of each equation above by the constant $r$.  This yields a system of nonlinear equations of degree 3.  If you want a system of nonlinear equations of degree 2, that is achievable too; translate the clause $(x_i \lor x_j \lor x_k)$ to the two equations

$$(1-x_i) (1-x_j) = 1-y_{i,j} \pmod m \qquad \qquad (1-y_{i,j})(1-x_k)=0 \pmod m$$

where $y_{i,j}$ is a new fresh variable (different for each clause).