What is happening to my car?

I was changing the battery on my 97 Ford Taurus with a 3 liter engine and it worked fine for a while. About a week later it didn't start and I had to recharge it. It worked for a week later and then the same thing happened, and I have no idea what happened. By the way the battery was from September 2016, and I don't think these things happen with a battery from this long ago.

Are you driving around with the red battery charge warning light on?
You need to charge the battery and then go to an auto parts store where they test batteries and charging systems for free. You probably have a bad alternator. You might want to check the fusible links in the engine compartment fuse box. One of them will blow to protect the electrical system and alternator if the battery is hooked up backwards while jumping it. The fusible links are the larger fuses.

You can use a multimeter usually found at local auto part stores, and test your alternator. It sounds like your battery is not being charged by the car, and your having to manually recharge it.
Set multimeter to DC car batteries run on DC power.
Find the positive cable on the alternator (usually a single red wire) and attach the (+) side of the meter to the red wire (red to red), and the other black wire (-) to the negative side of the battery (black side of battery), and see what your multimeter reads. It should be anywhere from 11.0-16.0 volts
If it shows nothing or anything under, your alternator is not charging your car battery, and you said it is a 1997, that is about the right time for it to go out.

Oh car MUST BE RUNNING to test alternator. If your alternator does show over 12.0 then you have a battery problem. Replace

Either defective battery or alternator. Charge it up and have it tested at an auto parts store for free. They will even charge it for you. They can even test your alternator when the car is running.

I have been through defective batteries and alternators. Let them check it with the proper tools. They will know what specs when it comes to your battery and alternator to tell you what is going on more than most people here can.

BTW, for a fact you can have a perfectly good battery reading 12.5 volts and STILL have a defective alternator.

You mean because maybe they were better quality? A battery, no matter how new, will be destroyed if it becomes completely flat. If you ran it way down and then left it over winter that would definitely kil it. For good. So in that case, maybe you have got 2 bad batteries. But there's another possibility which is that the alternator is not charging the battery. To determine this the voltage must be 13.9 with the car idling.

If you can run a car for a week after charging the battery, the battery is fine.
It is not charging. Do you have a light that comes on when you just turn the key without starting, which is a battery "warning" light? If it is there somewhere, it should extinguish if the engine is started.
If it's not there, you could have a broken wire to the alternator, the bulb in the light might be blown, or the alternator is unserviceable for some internal reason.
A voltmeter connected to the battery terminals will soon show if an alternator is working. It will read quite a bit more (13.5- 14.5V) than the battery's nominal 12 volts with the engine running, just after starting it. It can drop lower later as the battery recharges.
Switching on headlights etc. Should make no difference to the voltage reading, either.
Note that when you test an alternator this way, you should also rev.Up the engine a number of times, and watch the battery warning light does not flicker on. If it does, it usually means the field brushes of the alternator are worn down. These days, even to replace the brushes may not be possible for a handy person who might profess to understand cars.

Battery faults can occur in that time scale (but you would get warranty so you would hope that), other options are parasitic current draw (the car is drawing too much power from the battery when the car is turned off) or your alternator is faulty and not charging properly

The alternator is not providing sufficient charge to the battery

Most likely problem? Bad alternator.

The Mitsubishi Master Tech said it best:

"Are you driving around with the red battery charge warning light on?"

Well, ARE you?

Your car has lost it's memory.