As WALTER-MILANO-ITALY indicated above, it is possible to gain access to the hard disk on another computer using FireWire target disk mode (if both machines have FireWire ports). A FireWire cable would be needed for this. See the manual for your computer(s) and the article below for details.
http://support.apple.com/manuals/
http://support.apple.com/kb/ht1661
In order to remove a hard drive, you would have to disassemble the computer. This can be more or less difficult (a quick look at ifixit.com should give you an idea about this) depending upon which model it is.
Once the hard drive has been removed, you can either keep it, erase/use it with another computer, or mechanically destroy it. If you wish to connect it to another computer, something like this example could be useful:
http://www.newertech.com/products/usb3_universaldriveadap.php
You are not supposed to drag a hard disk with an active system to the Trash. Dragging individual files to the Trash (and emptying the Trash) is not going to be sufficient, unless the space is overwritten. It is not wise to recycle a computer without having properly erased or removed the hard disk drive.
Jan