Resistance
Resistance represents just what it sounds like: How strongly does an element "resist" the current flowing through it? If the resistance of a component or section of a circuit differs from what's expected, that's an indication of a problem. In addition, it's a good idea to start with resistance measurements, because they can often be taken when the circuit is not powered, making it safer to test.
Continuity
A special case of resistance measurement is continuity. A positive continuity measurement is essentially saying there is a short circuit between the two measurement points. This is useful for checking that a circuit has been built according to specification. It's also useful for detecting short circuits that shouldn't be there, as between positive and negative power connections.
Voltage
Voltage is also called "electro-motive force." It represents the amount of "push" behind an electrical current. Without sufficient voltage, motors don't turn and circuits don't run; while too much voltage can literally burn out components . Voltage can be either constant and steady, DC, or cyclically varying, AC. Most multimeters can handle both, but the internal measurement circuits are different, so you'll need to select the appropriate setting.
Current
Current measurements are different from voltage and resistance measurements, because voltage and resistance measurements can be made on the intact circuit, while a current measurement requires that the meter be inserted into the circuit. The circuit connection must be broken and the current routed through the meter instead of through the usual wire or other connection. Because the measurement is done differently, with much larger amounts of current than the other styles, the multimeter will have a special socket (or maybe even two) for making current measurements.
Range
A multimeter offers not just different styles of measurement, but different ranges as well. It's good practice to start with the meter set at the high end of the range, and move it down as necessary to get a more-accurate reading. For current, this may mean switching from a high-range to a low-range input for the test lead.