T-tests and chi-square tests are statistical tests, designed to test, and possibly reject, a null hypothesis. The null hypothesis is usually a statement that something is zero, or that something does not exist. For example, you could test the hypothesis that the difference between two means is zero, or you could test the hypothesis that there is no relationship between two variables.
A t-test requires two variables; one must be categorical and have exactly two levels, and the other must be quantitative and be estimable by a mean. For example, the two groups could be Republicans and Democrats, and the quantitative variable could be age.
A chi-square test requires categorical variables, usually only two, but each may have any number of levels. For example, the variables could be ethnic group — White, Black, Asian, American Indian/Alaskan Native, Native Hawaiian/Pacific Islander, other, multiracial; and presidential choice in 2008 — (Obama, McCain, other, did not vote).