The predicted value, represented as the Y with a hat, is Ys expected value at its observed X value for a given unit in the sample.
Remember: the "i" designates a unit in the sample.
The observed value, represented as the Y without a hat, is the value of Y we measured or collected for a given unit in the sample.
The residual, represented as the e, is the difference between the observed value of Y and the predicted value of a Y for a given unit in the sample.
The beta or, in our case, the regression coefficient is represented by the B. It represents the expected change in Y given a 1 unit increase in X.
The alpha or intercept is represented by the a and represents the expected value of Y when X is equal to 0.
Beta and Alpha are discussed more in the interpretation section.
TL;DR: Regression predicts the value of Y and tells us for every observation what we expect. However, we rarely observe exactly what we expected. The residual is the difference between what we observed and what we expected to observe.