You MUST always unplug the PS or otherwise remove the AC power to it whenever you make any connections or component changes inside the computer case. You can easily damage the PS or the mboard or anything connected to it if you don't do that.
ATX mboards are always being powered by ATX PSs in some places even when the computer is not running as long as the PS is receiving live AC.
It's a huge risk to use an el-cheapo PS. They are a lot more likely to damage the mboard or anything connected to it when they fail.
"...accidentally plugged in the power plugs backwards into the cd and dvd drives"
How did you manage that? The shape of the power connectors plastic shroud fit in the socket on the drives only one way - it is quite difficult to force them in upside down.
In any case, the DVD drive should eject or retract it's tray when you press it's button regardless of whether the data cable is connect when the computer is running.
It has also probably been instantly trashed by you plugging in the power connector backwards, if not right away, the instant you tried to boot the computer after that, despite the fact you noticed no smoke from it.
If you can't get the computer to boot, try it connected to another computer or try a known good PS connected to this mboard (the PS must have a significant load on it - e.g. at least the mboard - in order for the PS to start up).
"All the fans are running fine but the hard drive does nothing and the monitor says no signal even the power button LED doesn't light up"
That is either the symptoms of a damaged PS, more likely, or a damaged mboard.
Damaged PS often partially work but the mboard cannot fully boot.
Failing power supplies are common and can cause your symptoms.
Check your PS.
See response 4 in this:
http://www.computing.net/hardware/w...
Do you have another known good PS you can try?
You may have damaged your newest PS when you plugged in the power connectors backwards.
If a known good PS allows the computer to boot normally, you just need to replace the PS again.
If the system still won't boot you have probably trashed the mboard, or the cheap PS failing damaged the mboard but it took a little time to fail.
Usually the ram, cpu, and hard drive(s) are still fine if the mboard has failed. The cards in slots may or may not be damaged, but usually they aren't.