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Polynomial Regression

Is a model with higher-order polynomial terms still "linear regression"?

Linear Regression, in the sense of being "linear in the betas", can also include terms with higher orders, such as quadratic, cubic, or higher order terms.

How can I tell if a higher-order term might be useful in the model?

The need for polynomial regression can often be assessed by a review of the scatterplot of residuals vs. X in a simple, first order, linear regression model.  There may also be physical reasons to expect the need for polynomial terms in a model.

Hierarchical approach to fitting.

When including higher order terms in a model, a hierarchical approach to fitting is used where all lower order terms must also be kept in the model ... even if the lower order terms are not statistically significant.  This is contrary to the principle of parsimony, but is important for polynomial regression. 

From Kutner: "If a polynomial term is retained in the model, then all related terms of lower order are also retained."  (See Applied Linear Regression Models (Kutner) section 8.1 page 299)