I'm going to base this solution on two scenarios from the info you have posted.
1) The power to your home/business went out. Now it's back on.
2) The power has been on all along, and the computer quit.
Based on 1) above;
The Power On switch could be bad, or the Power Supply.
A voltage spike could have occurred, and taken out the ATX power on switch.
(A generic ATX Power On switch is around $5. Example,http://www.axiontech.com/prdt.php?item=78201
A voltage spike could have taken out a power supply that is of low quality, and/or on it's last leg.
The power supply used in an HP Pavilion a1100y desktop computer is of low quality.
Low quality components are used inside.
(Bestec, and Delta are two of the low quality power supply's used by HP in this time frame)
Plus add to the fact that the computer has some age on it, (5 years old?), and could be dirty inside.
Dirt,dust, etc. 'kills' computers and power supply's.
There is a simple test to perform to see whether the problem is the Power Supply, or the Power On switch.
(The Power On switch is located within the plastic Power On button assembly)
If you wish to know how to perform this test please post in a Comment.
Based on 2) above;
The Power Supply is bad.
The test would be either to use a multimeter, and check the three main voltages coming out of the Power Supply,
Substitute the power supply with a KNOWN to be good compatible unit.
(250 to 300 Watt)
(Sometimes access to another known to be good, compatible power supply may be obtained from a computer that is no longer in use. The user has upgraded to a better computer.
Compatible being that the Power Supply has the required Wattage, and required power cables)
[The three main voltages produced by an SMPS (Switched-Mode Power Supply) used by the Pavilion a1100y desktop computer are;
A) 3.3 Volts DC (Orange wire)
B) 5 Volts DC (Red wire)
C) 12 Volts DC (Yellow wire)
In comparison two D cell flashlight batteries produce 3 Volts DC ]
This is HP Support>Pavilion a1100y desktop computer> Motherboard Specifications,http://h10025.www1.hp.com/ewfrf/wc/document?docname=c00499325&tmp_task=prodinfoCategory&lc=en&dlc=en&cc=us&site=null&lang=en&key=null&product=1118244
Looking at the motherboard illustration, and photo at the bottom of the page, we can see what power cables are needed for the motherboard;
A) 24-pin ATX main power cable
(Connector on motherboard illustration marked as ATXPWER1. In the photo it is whitish, and has two vertical columns of 12 square socket holes. It is to the Right of the Ram Memory slots)
B) 4-pin ATX main power cable.
(Connector on motherboard illustration is marked ATX12V. It is to the left of the Intel Northbridge chipset - Intel 915GV GMCH.
In the photo it is the whitish square connector with four square socket holes. It is to the Left of the
aluminum finned Heatsink, sitting on top of the Intel Northbridge chipset)
Other power cables needed are;
1) A SATA power cable for the SATA harddrive
2) One or two 4-pin Peripheral (Standard) power cables for the optical drive/s. (CD/DVD drives)
The Processor fan cable attaches to the motherboard. (CPU_FAN)
The computer case fan attaches to the motherboard. (SYS_FAN)
The ATX form factor is used primarily for describing a motherboard.
Form Factor denotes the size, and shape, and power cables for a Power Supply.
The ATX form factor for a Power Supply;
Size: Approximately 6 inches Wide, 5-1/2 inches Long, and 3-1/2 inches Tall.
A $40 ATX power supply with 300 Watts will do just fine. No need to buy an expensive Power Supply based on advertising as a direct substitute.
One example of an inexpensive reliable ATX power supply for a Pavilion a1100y desktop computer,http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=1389575&CatId=1078
Has all the power cables you'll need. Actual maximum wattage rating is 350 Watts.
For additional questions or to clarify anything I have stated, please post in a Comment.