When you turn the key you hear a click under the hood, but the engine doesn't turn over. The battery is brand new.

Check the battery connections, the connection to the starter, and the starter itself.

If the starter motor doesn't turn, but the starter solenoid (relay) "clicks" when the ignition switch is turned to Start, the problem is with the battery, starter solenoid, or starter motor connections. Remove and fully charge the battery. Even new batteries should be charged, prior to installation. Most auto parts stores will do this for free. Once you establish that the battery is, in fact, good, clean the battery terminals and connectors, and ensure the connectors are tight. To check the solenoid, put the transmission in Park and remove the push-on connector from the solenoid “S” terminal. Connect a jumper lead (12 AWG), or preferably a remote starter switch, between the positive battery terminal and the exposed “S” terminal on the solenoid. If the starter motor now operates, the solenoid is fine and the problem is with the ignition switch, neutral start switch, or wiring. If the starter motor still does not operate, replace the solenoid.