For laptop: Toshiba Satellite A105: My battery won't charge/computer won't turn on when plugged in. No lights show when plugged in. Don't know if it's a problem with the chord or the computer. Don't want to buy a new power chord until I know.

Start by trying another power outlet.

The power adapter should have its own light. If it doesn't light up then the adaper is bad. Kensington.com sells very inexpensive universal power adapters. You might also find a good deal on eBay.

If the adapter lights up then it may be your battery. Notebook batteries are only good for 2-3 years. If you let them sit for a long time without a charge they may only last 1-2 years.

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Did these people even listen? if you aren't getting any lights on the computer when it's plugged in, either your motherboard or poweradapter is dead. Do try removing the battery and then plugging it in, but that probably won't help.
I don't know about yours, but my adapter for my x205 - sli doesn't have a power light, so maybe try to find one at a store that can be returned if it doesn't help.
For your computer, it looks like the connector that the adapter plugs into on the computer is not hard-wired into the mother board and can be replaced.
Unfortunately, if it is not the power supply, or connector, it is almost guaranteed to be the Mother Board. Replacements are usually much more expensive than the computer is worth for laptops. sorry :( This just happened to my laptop last night, which is why im trolling.

It's obviously that your toshiba satellite a105 battery is worn out. You need a new battery.
Battery is very weak to power off, so prepare a battery is necessary.
When choose a replacement Toshiba satellite a105 battery(original battery is too expensive for me), check the battery model carefully.