On some "direct drive" washing machines there is a hard rubber coupling between the motor and the drive shaft. This coupling is designed to disintegrate if the motor is overloaded, resulting in the exact symptoms you are describing.
It's a $10 part, but you really need a service manual for your washing machine to do the installation. You can get these service manuals from your manufacturer's service center (we have a Sears Service center down the street from my house). They are pretty cheap, but you have to ask for them by model - they usually don't put them out on the shop floor. You MIGHT be able to figure things out without the service manual, but you run a risk of screwing up the machine, and your repair time will at least double, in my experience.
The repair itself is relatively straightforward, but I highly recommend getting your service manual before taking things apart. There are some areas of the machine that retain a voltage even when the machine is powered off, so you don't want to get zapped. Also, some parts of the motor assembly are sharp and/or very heavy, and can easily injure if you have your hands in the wrong place during dissassembly.
Some basic tools, and a few hours of your time are all you really need to save hundreds in repair costs. If you can build a computer, a washer repair should be a piece of cake.
If you need any Sears service manuals, I'd be happy to pick one up for you at the service center nearby - send me a PM.
Good luck!