This ambiguous term has two different meanings.
1) A model is said to be "linear in the variables" if the variables in it appear only in linear combinations.
2) A model is said "linear in the parameters" if its parameters appear only in linear expressions. A polynom of degree higher than one is linear in the parameters, but non linear in the variables.
A Neural network is linear neither in the variables, nor in the parameters.
In very general terms, models that are linear in the
parameters are simpler, but cannot generalize
nearly as well as more sophisticated model (e.g. Neural
networks) models that are not linear in the parameters.