A scatter plot consists of points where one of the coordinates all the points represents one variable and the other coordinate represent another variable.
If the scatter plot is not linear but curvilinear in nature, it shows that the two variables that are being analyzed do not have a linear relation. For a unit change in one variable, the change in the other variable is not constant and independent of the two points that are being considered. Instead, the change depends on the points that are being looked at. The mathematical relation between the two variables cannot be expressed as a simple linear relation. It is a more complex relation that has terms with powers higher than 1.
For example if the increase or decrease in mental abilities is being plotted against the age, there is large difference in the rate at which the two change. It is not possible in this case to use a linear relation to estimate the change in mental ability given a particular change in age.