My power went out and when it came back on the computer would not come on

EMachines uses low quality hardware components inside their computers. This is the way they can sell them at budget prices.

One main hardware component is the low quality power supply they use.
A generic power supply made by Bestec. (A generic power supply manufacturer)

(PSU. Power Supply Unit. The type of PSU used is a SMPS.
Switched-Mode Power Supply,

http://en.wikipedia.org/wiki/Switched-mode_power_supply )

Low quality;
A) Electrolytic Capacitors (Radial Aluminum Electrolytic Capacitor)
B) Rectifier Bridge
C) MOSFET/s
D) Wiring is too small in gauge
to name a few of the components.

However IMHO it is also best to check out the Power On switch.
There is a simple test to see whether the problem is a faulty Power On switch, or a faulty Power Supply.

The test involves bypassing the Power On switch. The Power On switch is not involved in this test.
A jumper wire is used on the Soft Power On wire, [ PS-ON ], and a Ground wire in the ATX main power cable connector.

If you bypass, and the Power Supply comes on you have a bad Power On switch.
If the Power Supply does Not come on you have a bad Power Supply.
If you would like to know more in detail on how to perform this simple test, simply post so in a Comment.

The Power Supply used is just a common ATX version.
ATX refers to the Form Factor of the power supply.
Regular size ATX form factor is approximately 6 inches Wide, 5-1/2 inches Long, and 3-1/2 inches Tall. (Height)

There are also differences in the pinout of the ATX main power cable.
Yours may have 20-pins, or 24-pins.
Power Supply's sold now take this into consideration.
Their ATX main power cable can be used as a 20-pin, or a 24-pin version.

[ General example of a 20-pin ATX main power cable, and it's connector,

http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain20

Example of a 24-pin ATX main power cable, and it's connector,

http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain24

Your motherboard may also require an additional power cable, commonly referred to as a P4 power cable, or P4 ATX 12V power cable, or just ATX 12V.
The actual name is 4-pin ATX +12 Volt power cable,

http://www.playtool.com/pages/psuconnectors/connectors.html#atx12v4

If you would like help in finding an inexpensive reliable Power Supply, please post in a Comment.
I will need to know what Model Number of eMachine you have. It's on the back next to the Windows product key, or up on the side of the computer tower.

If you need anything explained that I have stated here, please let me know in a Comment.
Regards,
joecoolvette