PC won't power down all the way

ah what exactly do you mean modem and router shows activity? That the NIC still has a link?

some new mainboards with built in NIC cards have a Wake on LAN feature that even if the computer is powered down the NIC is active so you can remotely wake the computer. I have this on all my workstations so I can use a nifty little program called Magic Packet [AMD PC Net Family program] to wake em all up. I pull down the servers and all the worstations once a month to clean em out of dust bunnies, hairballs, the occasional errant cockroachs, [who seem to love power supplies] spiders, and whatever else seems to find their way into the workstations. [still can't figure out if someone put the curly pieces of french fries in a workstation or if it was the cockroaches]. Then send a network wide wakeup call.

See if you have Wake-On-Lan enabled in your BIOS

To add to what D.R. gave you:

First, please list your hardware.
There are MANY hardware specific answers to many problems.

It just takes too long to type in all the possible answers.

If this is a Socket A system, make sure that the CPU bus speed and multiplier are set properly.
I have seen MANY strange power problems on Socket A systems that have been corrected by changing the CPU speed settings.

As far as immediate power up when you turn on the power supply, check the power state in BIOS power management settings.
If you want the PC to stay off, you need to have the setting to always off when power is restored.

Power management may also be set to go into hibernate rather than shut down.

Keep in mind that for ANY ATX system, when you shut the PC down, it is NEVER fully off.
There is always 5 volt standby power supplied to the motherboard.

Chas