My macbook late 2008 (aluminum unibody) has stopped charging

Yoiu probably wrapped the cord to death. When I wrapped my cord around the magsafe it got tighter and tigher untl the wires just unplugged themselves on the inside. It is better to have something to stor it in and keep them lose. Just take it to the store as soon as they plug it in with one of theirs that works they should know that you just need a new magsafe power adapter. they cost a lot to replace and they break way too easy, so be nice to it or you have to buy them more often.

Hello,

Also, it does say that my battery needs to be serviced, but so far i have noticed no other issues. Suggestions?

Get a new battery.

I have heard that there is a wrong and write way to wrap up the cord. If there is, what is it?

Mac notebooks: Reducing cable strain on your MagSafe power ...

If you don't live near one you can always just order one online.
Just figure out which one you have:

http://store.apple.com/us/mac/mac-accessories/power#!

You can get a new third party 60W power adapter on eBay for under $20.
http://www.ebay.com/sch/i.html?_nkw=macbook+power+adapter&_sacat=0&_sop=12&_dmpt =Apple_Laptops&_odkw=macbook+power+adapter&_osacat=111422&_trksid=p3286.c0.m270. l1313

Or for $80 from Apple http://store.apple.com/us/product/MC461LL/A