EMachine Model T3828 No power
Greetings! As the first answer suggests, you must be comfortable taking things apart. I've actually just finished diagnosing the same problem on a customer's computer, and Here's what I did.
First, I remove the power plug, open up the computer, and inspect the caps (capacitors). If any of them are 'leaking' or 'bulging' or becoming 'pointy', then you are done, your motherboard is at fault for certain. If not, then I continue ...
Second, I have a power supply tester, something you can get at most small mom's-n-pop's computer stores, or online at newegg.com or tigerdirect.com for as little as $25 to $50. I unplug the power supply from the motherboard, both the 20-24 pin connector and the 4-8 pin connector (the 12 volt supply on newer systems 4-5 years old or newer). I plug these into the tester, and test the power supply by plugging the power back into the power supply. If all of the lights on this tester come on, then the power supply works. If they don't, or if they do, but you can smell a rancid burn't smell coming from the power supply when you put your nose next to it, then the power supply is shot. This would be a good time to have another power supply to use to test with the current setup, simply setting atop the current power supply, and replacing the plugs so that the working test power supply is plugged into everything instead, and if that powers the system and it turns on, then you know the power supply is the faulty part.
Next, if the power supply is good, or the test power supply doesn't turn the system on, then I continue to testing the motherboard. I have a POST tester to use for this, but you may not want to spend the $50 on this, and it's rather complicated, so now remove everything from the motherboard except the CPU, RAM, and HDD. this means remove any NIC's, Modems, CD drives, anything and everything. Now try with the test power supply connected to power the machine. If it starts, then it's one of the other parts. if it doesn't, then remove the HDD. Try again. If nothing, then remove the RAM and try again.
If nothing, then the last resort is to remove the CPU heatsink/fan, inspect the CPU, and remove and re-seat the CPU. This step in my opinion should require at least some prior experience with CPU's, and I would suggest having a can of CPU cleaner and a tube of CPU thermal grease to replace the now exposed thermal grease. The reason I would suggest prior CPU experience, is that the thermal grease is 110% necesary for the heat to transfer to the heatsink, and while others may disagree, there is a too much, and it's not a good thing. You need only a very small thin layer of thermal grease, only around 100th of an inch thick, even thinner.
So once re-seating the CPU, re-connect the heatsink and cpu, and try again. Then begin connecting parts one at a time, starting with the RAM, then the HDD, then the optical drive, etc, until either it starts, or everything's plugged in and it doesn't. Sometimes in odd situations, the system will not start with just the CPU, RAM, and HDD, while it will if you add the NIC or the Optical drive. In these strange cases, there's usually something wrong with the motherboard, cpu, or ram, and you'll want to find out from a profesional, or simply replace the motherboard.
In this most recent case I diagnosed with my customer's computer, the motherboard was faulty, with no visible evidence of damage. E-Machine computers most common problem are the motherboard and the power supply. In my experience with my own company, and with the last two computer shops I worked with, E-Machine's have these problems usually at least once within the first year, covered under the Manufacturer Warranty, and then once again after about 5-6 months past the first year. This is due to the cheaper quality parts, cheaper quality soldering, and cheaper construction in general of E-Machines. E-Machine can produce a much less expensive product, though this becomes evident in the quality and expected lifetime of the product.
Sorry to write a novel, but I hope this helps give some perspective on what is done by a technician. Realize, As I work on diagnosing a problem, I'm thinking of a lot more things than just what I've written here, this is just a general synopsis of what I'm doing when I diagnose a system, and should just be used for general diagnostics in the home, or as a begginers basis for working out problems.