I actually work at Maxim... besides the LVDS digital video switches DrFriedParts mentioned, we also have crosspoint switches and multiplexers for NTSC/PAL standard definition analog video signals: http://www.maximintegrated.com/en/products/analog/video-products/crosspoint-switches-and-multiplexers.html
crosspoint ("any-to-any") means input1 can be connected to output1, or output2, or output3, or output4. And at the same time, input2 can be connected to output1, or output2, or output3 or output4. And so on. These are more expensive and more complicated to command than a simple multiplexer, because each output needs to know "on or off" but also "which input", so at least three bits x 4 channels = 12 control bits, maybe more. Example: Maxim Integrated MAX4359 http://www.maximintegrated.com/en/products/analog/analog-switches-multiplexers/MAX4359.html (this is an active buffer). This one uses 7 pins: A2,A1,A0 select one of the channels, and D3,D2,D1,D0 determine how to configure that selected channel. (From your comment to @DrFriedParts I think this probably is not what you need.)
MUX chooses one output between two or more inputs. A 2:1 mux selects from 2 possible inputs to drive 1 output. Example: Maxim Integrated MAX4025 http://www.maximintegrated.com/en/products/analog/video-products/MAX4025.html (this is an active buffer). These are less expensive and simpler to control than a full crossbar, because each output only needs 2 bits: A/B selects between "input A" and "input B" for that output, and EN enables the output driver. So OUT1 is driven from either IN1A or IN1B, and OUT2 is driven from either IN2A or IN2B. You could externally connect IN1A and IN2A to the same signal source, or not.
SPST switch (single-pole, double-throw) is just one on-or-off switch. So input1 can connect only to output1 through one switch, and input2 can connect only to output2 through another switch. These are less expensive and easier to control, because each switch just needs to know "on or off", one bit for each switch.
For any of these, you will need a custom PCB (printed-circuit board) layout -- video signals are high bandwidth (between 200MHz and 600MHz depending on format), this type of signal will not work very reliably on a solderless breadboard. Video signals also require correct output termination and controlled-impedance traces, to avoid distortion.
If you really want to try prototyping this on solderless breadboard using old-fashioned DIP parts, your best bet is the old
74HC4066 quad SPST analog switch. I'm not sure whether or not this device has enough bandwidth to pass an NTSC/PAL video signal -- typical bandwidth is only about 100MHz, and standard-definition video really wants several times more bandwidth. The capacitance of the solderless breadboard will also reduce this bandwidth further, maybe as low as 10-20MHz.
This question really seems to be more about video switching not specific to arduino, it might be better to remove the arduino tag to avoid being migrated to the arduino stackexchange.