This statistic serves three functions. It can test whether the means or averages of two different groups on one variable is the result of chance, such as the grades of two different sections of a class; whether a mean is statistically different than some standard, such as whether the average lawyer makes more than $100,000 a year; or whether the same group has a different mean on different variables, such as the grades of the same group of students in different classes.
The t-statistic has close to a normal distribution, which is shaped like a bell curve. A normal distribution, by convention, is assumed to have a mean of zero and a standard deviation of one. A t-score can fall anywhere on this distribution. If the t-score is far from the mean, then the relationship is probably not due to chance.
Alpha is the probability that the relationship is due to chance. If the alpha value chosen is 0.05, that means that the t-score is in the extreme 5 percent of the distribution and that there is only a 5 percent probability that it is due to chance.