In the context of calculating standard deviation Wikipedia says that the residual is the difference between an observation and the mean.
Is the actual term "residual" referring to an observed difference (which could be negative), or is it referring to the difference after it is squared and then square rooted (to make it positive)? If only one of these is the residual, what is the other called?
Residuals can be both positive or negative. In fact, there are many types of residuals, which are used for different purposes. The most common residuals are often examined to see if there is structure in the data that the model has missed, or if there is non-constant error variance (heteroscedasticity). However, the absolute values of the residuals can also be helpful for these purposes. To see some examples, it may help you to read my answer here: What does having constant variance in a linear regression model mean? In the figures at the bottom, look at the bottom two rows. The middle row shows typical residuals and the bottom row shows the (square root of the) absolute values of the residuals.
