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An 'edge connector' is the portion of a printed circuit board consisting of traces leading to the edge of the board that are intended to plug into a matching socket.
An 'edge connector socket', often popularly referenced simply as a 'slot', is any type of female electrical connector for use with printed circuit boards having matching edge connectors. They consist of a plastic "box" open on one side, with pins on one or both side(s) of the longer edges, sprung to push into the middle of the open center. Connectors are often keyed to ensure the correct polarity, and may contain bumps or notches both for polarity and to ensure that the wrong type of device is not inserted.
The opposide side of the socket is often an insulation-piercing connector which is clamped onto a ribbon cable. This can be seen inside a personal computer as the connection to a floppy drive. (5.25" floppy drives actually used an edge connector, while 3.5" drives use a pin connector instead, but both typically use the same insulation-piercing connector to the ribbon cable on the back of the connector.) The other side may also be soldered to a motherboard or daughtercard. This is seen inside every PC for connecting expansion cards and computer memory to the system bus. Slot 1 and Slot A were also commonly-used at one point, the processor being mounted on a card with an edge connector, instead of directly to the motherboard as before and since.
The edge connector is a money-saving device because it only requires a single discreet female connector (the male connector is formed out of the edge of the PCB,) and they also tend to be fairly robust and durable. For a time they were used in the vast majority of connectors found in computers, but modern computers have demanded many more pins than can easily be accommodated on the edge of a reasonable size board, and today more traditional male/female connectors are more common.