Aztek will not start.

If it comes to having to buy a starter, I'd recommend a new starter over a used one.

As for the *click* then nothing that's been happening... if the battery is bad, you would hear a *tick-tick-tick-tick-tick* sound. That's the starter solenoid magnetizing and hitting the starter connection. If you hear just one single *click*, I'm guessing you either have a bad starter or a bad connection.

You either have a defective battery or the battery cable or cables have acid intrusion that has "wicked" inside the cable and this causes a bad connection, the only fix is to replace the cable or cables, this problem is mostly limited to the positive battery cable because of the current flow direction. So 1st have the battery tested and see what it shows, almost any of the many discount auto parts stores will test the battery for free.

For tonight I would just unhook that battery, and call them in the morning. I wouldn't do anything other then unhook the negative battery cable. You don't want to do anything to void the warranty.

Here's a video showing you how to remove battery. Battery Replacement 2001 2005 Pontiac Aztek 2003 Pontiac Aztek 3 4L V6

a dead or shot battery.

A rapid clicking noise in a GM starter is a sign of insufficient power going to the battery. Make sure all your connections have bare, shiny metal to metal contact. Do a "load test" on the battery. It may show enough power on a voltmeter only to fail when it is forced to produce it. On a load test it shouldn't drop below 10.5 -12 volts and it should bounce back to near 14 when the load is off. Check the ground strap between the engine and body and the ground at the battery.
If everything is good I would replace the power cable from the battery to the starter. I've had a few of them that looked good but the car wouldn't start until I replaced the cable. Hope this helps.

You said you tightened the battery cables, did you check the other end of the negative where it bolts to the engine. When you tightened the cables you also used a battery brush and cleaned the posts and ends?

earth cable from body to engine and battery to chassis ,clean the connections and fit an additional cable ,also clean and grease the battery terminal

If the car is clicking, it is most likely the starter engaging, but it doesnt have enough power to crank. Charge the battery or try getting a jump.