DO NOT OPEN A POWER SUPPLY!!!!
Inside an SMPS, (Switched Mode Power Supply), are Electrolytic Capacitors.http://en.wikipedia.org/wiki/Switched-mode_power_supplyhttp://en.wikipedia.org/wiki/File:ATX_power_supply_interior-1000px_transparent.pnghttp://en.wikipedia.org/wiki/File:Capacitors_electrolytic.jpg
(Radial. The bottom capacitor in the photo)
Electrolytic Capacitors slowly build up a charge, then release it all at once.
You can compare it to a large swimming pool being filled up with a garden hose, and then one wall of the pool being taken down all at once.
At the bottom, of the Radial (Style) Aluminum Electrolytic Capacitor shown in the above photo link, are two leads.
One is Positive, the other is Negative.
Touch your finger/s to the terminals on the bottom of one of these babies, and the shock can be BAD to FATAL!
If your fingers touch, and complete a circuit that one, or more capacitors are in, the shock can be BAD to FATAL!
Don't open that power supply!
Besides, a fuse goes out All At Once.
There is no work a little bit, then turn off.
It just goes Poof!
(The fusible element inside burns into two separate pieces)
"You press the power button, and it doesn't turn on all the time"
As has been suggested, I also concur that you have a bad power supply.
But you could also have a bad Power On switch, located inside that plastic Power On button.
This is an example of a Power On switch, that I have found fits a lot of computers,http://www.directron.com/atxswitch.html
I'd like you to get this switch, and install it.
From there the diagnosis will go to the Power Supply.
Also would like you to make sure the ram memory is seated tight, and the Processor fan is spinning, and looks to be spinning at the right speed.
If the fan spins a few times, and stops, then spins a few times, you know the fan has a bad bearing. Replace.
If it looks to be spinning too slowly also, you'll know it
Be sure to follow anti-static precautions before you reach inside your computer, and the computer is unplugged from power.
If you do not know what anti-static precautions are, I'll be glad to tell you. Simply state so in a Comment.
(Believe upper right of your page)
As for the Power Supply test, you need either a multimeter, or a power supply tester.
An inexpensive multimeter, that will work just fine for this application, ranges from $6 to $12. The Voltages are only 3.3 Volts, 5 Volts, and 12 Volts. (DC)
Two C cell flashlight batteries are 3 Volts. (DC)
Your computer power supply puts out three main Voltages, as stated above.
A) 3.3 Volts (3 point 3)
B) 5 Volts
C) 12 Volts
Orange insulated wires are 3.3 Volts
Red wires are 5 Volts
Yellow wires are 12 Volts
ANY Black insulated wire is a Ground wire.
This is DC Voltage.
There is a Positive, and a Negative.
The Positive (Red) probe lead of the multimeter touches a power wire. (3.3, 5, or 12)
The Negative (Black) probe lead touches ANY Ground wire. (Black)
IF, the 12 Volt power rail shows 11 to 13 Volts, the diagnosis indicates the Power Supply to be good.
(However read on)
IF the 12 Volt power rail shows 11 Volts, or less, the Power Supply is bad.
This is an example of a power supply tester,http://www.directron.com/pst03.html