Why is my power adapter not charging my computer?

If the computer is turning on when the charger is plugged in, it means the charger is most probably doing its part and the battery is dead. Just to make sure, jump to "second suspect" here below and test the voltage out. If the voltage out matches the rated output of the charging block (it is printed on the charging block, for example: VDC Out 19V or 20V), you confirmed that you need to try a new or known good battery.

If the computer is not turning on when the charger is plugged in, try the below to rule out power supply issues:

Tools: multimeter, eyes, brains.

First suspect: extension cable going from wall plug to charger. UNPLUG IT FROM THE WALL OUTLET then test. Either try a new one or test it for continuity across using a multimeter in continuity mode. Must beep across from where it is supposed to plug into the wall to where it is supposed to plug into the charging block. Must not beep across the parallel pins (say both pins that plug into the charging block).

Second suspect: charger itself if not working, or damage to the cable leading from the charger block to the laptop. Test at the tip of the charge cabled compare to the rated voltage output as noted on the block (usually 19~20V). Use multimeter in VDC mode.

If you suspect the cable is damaged, you can twist/bend slightly to see if this will restore continuity, and your options are to replace the block or cut the cable where damaged, reconnect, and insulate. If the block itself is bad, replace it. Don't trying opening it or fixing it. At least I wouldn't because of fire hazards or death by electrocution hazard. Not worth it.

Third suspect: the plug/connector inside the computer is damaged. Happens a lot. You need to take the laptop apart and examine that plug, and measure for continuity across along the red cable and along the black cable.

Finally, if you rule out the above, and are sure the laptop is not starting (vs. starting but not displaying an image), it could be that the board is damaged and not passing power to the battery or it is simply dead. There are numerous reasons that could cause this at board level. Not gonna list them here nor would I present to know them all.