Pearsons R is a basic statistical analysis that measures the strength and the direction of a correlation between two variables, such as income and weight or height and age. To use a Pearson's R analysis, data must be numerical, or interval, data. This includes age, temperature, length or other numeric data. As a test to see if your data are appropriate for a Pearson's R, ask yourself how many places after a decimal point you could figure the measurement. If it's two or more, you probably have good data for a Pearson's test. You cannot do a Pearson's R test for categories, such as "Freshman, sophomore, junior, senior" or "male/female."
If the variables in a research analysis are categories, then Chi Square is the test that will give you significant results. Categories include male/female, class in school, race, club membership and similar delineations. The Spearman's rank correlation is used for finding the strength and direction of correlations when at least one variable is an ordinal variable; that is, the variable categories can reasonably be ranked. This can include grades, teacher evaluation, or categories such as, "Agree, somewhat agree, not sure, somewhat disagree, disagree." Any of the correlation tests can only show a correlation, or a relationship, between two variables. None can indicate that one variable causes another to be true.
A t-test is the most common statistical test in research papers. It evaluates the differences in the means of two groups. For example, common experiments analyze the differences between a group that was exposed to a particular factor and a group that was not exposed -- the control group. A t-test may show the differences between test scores of at-risk students who were given tutoring and those who were not. It may evaluate the differences between the recovery days of patients who were given a new drug and those who received an old drug. This test needs one independent variable: a variable that groups people into categories, such as at-risk kids with no tutoring and at-risk kids with tutoring. The test also requires a dependent variable such as the test scores.
ANOVA stands for Analysis of Variance, and it compares the means of more than two different groups. For example, extending the test between tutored and not-tutored children, you could add a group of average performing children for a total of three groups. In the medical test, you can compare groups receiving no drugs, the old drug and the new drug. A test using ANOVA would still need an independent variable, but there can be three or more categories in this variable. The test also requires a dependent variable such as days of recovery.