Summarize:

The manufacturer's part number should be listed on the case of the relay. Look up the applicable data sheet and determine the voltage and current requirements of the control coil. This also may be printed on the case of larger relays. A diode around the pole is often used to protect the logic circuitry from damage due to noise spikes. The diode will be shown on drawings as a triangle with a bar across one corner of the triangle. The bar will be connected to the input, or positive connection, of the control coil. This will also be available from the manufacturer's data sheet, or may be printed on the case of larger relays. Relays may have one or more poles, indicated in drawings by a single line switch connected to a pin of the relay.  Each pole may have a normally open (NO) and or normally closed (NC) contact. The drawings will indicate these contacts as connections with a pin on the relay.  The relay drawings will show each pole as either touching the pin, indicating a NC contact, or not touching the pin, indicating a NO contact. Use a digital multimeter (DMM) to test the resistance between each pole of the relay and the corresponding NC and NO contacts for that pole. All NC contacts should read 0 ohms to the corresponding pole. All NO contacts should read infinite resistance to the corresponding pole. Use an independent voltage source appropriate for the rating of the relay coil. If the relay coil is diode protected, make sure that the independent voltage source is connected with the proper polarity. Listen for a click when the relay is energized. Use a digital multimeter (DMM) to test the resistance between each pole of the relay and the corresponding NC and NO contacts for that pole. All NC contacts should read infinite resistance to the corresponding pole. All NO contacts should read 0 ohms to the corresponding pole.
Determine the coil requirements of the relay. Find out if the control coil is diode protected. Assess the contact configuration of the relay. Test the de-energized condition of the relay contacts. Energize the relay. Check the energized condition of the relay contacts.