What does "curvilinear" mean? As far as I can tell, curvilinear is defined vaguely but means the same as nonlinear. Is that correct? Or does curvilinear have a distinct definition?
 A: To me, in the context of data analysis, it is always linked with the idea of leaning a topographic mapping of the data, so that samples that are mapped close by, are similar in a given sense. The wikipedia site on Nonlinear dimensionality reduction offers a nice overview. THe paper Laplacian eigenmaps and Spectral Techniques for Embedding and Clustering contains a nice description of a framework where the idea of manifold learning is linked with differential geometry.
In other words, curvilinear is to me related to the problem of learning a distance metric from data. The hypothesis is that the data lies in a smooth, low dimensional manifold. That learned metric correspond to the metric tensor as in the classical sense of the term.
A: "Nonlinear" has many meanings, only some of which are (directly) about curves. I would say that I have encountered "curvilinear" to mean smooth curves. So a parabola or a logarithmic curve are "curvilinear," but a bent line (e.g. from a simple threshold or saturation model, "broken stick" model, etc.) are not.
Caveat emptor: word use will vary by context. For example straight lines are themselves a kind of "curve" in some contexts. As always, if there is a specific usage of the word "curvilinear" that you are wondering about, a quote and citation or two would be helpful.
A: The lack of clear and consistent terminology is one of my pet peeves, but I don't see how there is any real solution.  For what it's worth, I often use certain words in a vague and hand wavy way to get at general ideas when I don't want to take on all the baggage of technically defined terms (e.g., "variability" instead of variance).  I have used "curvilinear" similarly.  I do like @Alexis' description.  If you wanted a more precisely defined version, I might posit that rectilinear would be a smooth function where the second derivative is $0$ everywhere, and that curvilinear is a smooth function where the second derivative is not $0$ everywhere.  
I do want to note that "curvilinear" and non-linear should not be considered synonymous in statistics.  In statistics (e.g., regression modeling) "linear" is shorthand for linear in the parameters.  That is, all the parameters being estimated enter into the model as coefficients.  On the other hand, "non-linear" means that the estimated parameters do not all enter into the model as coefficients.  There are many cases where a function looks 'curvilinear' but is not non-linear (e.g., adding a squared term to a regression model).  This is a subtle point and it trips up a lot of students, so it is worth always stating explicitly.  For more on how a model that looks 'curvilinear' can still be a linear model, it may help to read my answer here:  Why is polynomial regression considered a special case of multiple linear regression? 
A: 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. 
