Nonlinear regression concerns models that are inherently nonlinear: that is, they cannot be expressed as a linear combination of parameters $\beta$. It is practically the same thing to say that a nonlinear model cannot be put into the form $Y = X\beta + \epsilon$ after a preliminary mathematical re-expression of $X$, $Y$, or both. For example, $Y = \log(X)\beta + \epsilon$ and $Y = \exp(X\beta + \epsilon)$ are both linear whereas $Y = exp(X\beta) + \epsilon$ and $Y = \log(X + \beta) + \epsilon$ are nonlinear.
(As usual, $Y$ is a dependent variable (or vector thereof), $X$ is a vector of independent variables, $\beta$ is a set of parameters to be estimated, and $\epsilon$ is random "error" with zero mean.)