Bed Slide tends to be the older name for a Slide Plate, the flat plate or plates on the bed which allow access to the shuttle or bobbin and other parts of the stitch forming mechanism.
Machines such as New Home usually have the serial number stamped on the slide plate, while a National machine will usually have the serial number on the rim of the bed, under the front slide plate. German machines often have a narrow slit in the slide plate acting as a spring to prevent the plate from accidentally falling out.
To remove the slide plate on a Singer 66 and 99 lift the right end of the plate and pull to the right. To replace the plate hold it and push from right to left, ensuring that the end of the spring on the left of the bed enters the grooves on the underneath of the slide plate.
Although not strictly a slide plate, some machines have a flip up plate over the bobbin area, notably Asian-made machines or some rotary machines e.g. White.