A Curvilinear Relationship is a type of relationship between two variables where as one variable increases, so does the other variable, but only up to a certain point, after which, as one variable continues to increase, the other decreases. If you were to graph this kind of curvilinear relationship, you will come up with an inverted-U. The other type of curvilinear relationship is one where as one variable increases, the other decreases up to a certain point, after which, both variables increase together. This will give you a U-shaped curve.
An example of a curvilinear relationship would be staff cheerfulness and customer satisfaction. The more cheerful a service staff is, the higher the customer satisfaction, but only up to a certain point. When a service staff is too cheerful, it might be perceived by customers as fake or annoying, bringing down their satisfaction level.