Problem with basic understanding of polynomial regression
"Linear regression" (aka. "ordinary least squares", OLS) refers to the type of estimator. Linear here means that you minimise the sum of squared residuals for a given (linear additive) regression equation. You can write a simple model:

$$y = \beta_0 + \beta_1 x_1 + u. $$ 

This would fit a linear function with intercept $\beta_0$ and slope $\beta_1$. So the "fitted" function would be:

$$ f(x) = \beta_0 + \beta_1 x. $$

You can "propose" basically any functional form to be fitted by the linear model, e.g. with a squared term (just add $x^2$ as feature):

$$y = \beta_0 + \beta_1 x_1 + \beta_2 x^2_1 + u. $$ 

This will give you the (estimated) quadratic function:

$$ f(x) = \beta_0 + \beta_1 x + \beta_2 x^2. $$

In essence, you can add any kind of linear transformation of $x$, e.g. $log(x)$, $exp(x)$ etc.

"Non-linear" models (there is no really clear definition), would be e.g. logistic regresion, non-parametric regression, tree-based models, neural nets etc. While each coefficient $\beta$ directly gives you a "marginal effect" in linear (OLS) regression, this is not the case for the other (non-linear) models. 

When you use regularisation in linear models, you simply add a penalty term to the loss function, so you minimise the sum of squared residuals given a penalty term. However, regarding your last question, I'm not quite sure what your actual question is.