Non-linear models are similar to linear regression models, except that the lines aren't straight.
Well, that's overly simplistic and a little tongue-in-cheek, but it does have a grain of truth. We're looking to find a formula that best fits the data, but without the restriction that the formula should be linear. This introduces a lot of complications and makes the problem significantly more difficult. Unlike linear regressions, fitting non-linear models typically involves a lot more guessing, and trial and error. Also, the interpretation of the model is much trickier. Interpreting a line is straightforward enough, but trying to figure out relationships when one involves a curve is much more difficult.