A line is described by a simple equation that calculates Y from X, slope and intercept. The purpose of linear regression is to find values for the slope and intercept that define the line that comes closest to the data. More precisely, it finds the line that minimizes the sum of the square of the vertical distances of the points from the line. Put the data in, and the answers come out. There is no chance for ambiguity.
Nonlinear regression is more general.
It can fit data to any equation that defines Y as a function of X and one or more variables. It finds the values of those variables that generate the curve that comes closest to the data. More precisely, the goal is to minimize the sum of the squares of the vertical distances of the points from the curve. Instead nonlinear regression requires an iterative approach. Start with an initial estimated value for each variable in the equation.