This may be hard to explain, but here goes:
I need to make a machine that has 24 BNC video inputs to 24 BNC outputs. These need to be selectable.
For instance, in the power up state, all inputs will go to all outputs respectively meaning that input 1 will go to output 1, input 2 will go to output 2, and so on.
The kicker come when I want to select which outputs that the inputs need to go to. For instance, on the fly, I want input 1 to be on output 6 (Purely for an example). I don't think I need help with the code at all, but need help selecting the components. I have already selected a Mega for this project. I plan on using a LCD screen with a keypad to do the selecting of inputs to outputs. I just cannot wrap my head around how to start with this and what devices (weather it be relays or ICs) to go between the inputs, arduino(s), outputs. . . Any thoughts? Please let me know if you require any further information on the topic.
What additional hardware are you looking at. I do not know of an Arduino that will do what you want. What bandwidth and frequency are you looking at. What frame rate. Check this link: What is Video Bandwidth ? 720p, 1080p, GB Transfer Explained - VdoCipher Blog
Seems like you want the ability to make 24x24=576 possible connections. That will be 576 individual relays, 24 fantastic 1x24 analog multiplexers, or many more feasible 8 channel analog multiplexers and all their connections.
Then BNC connections can be either 50 Ohm or 75Ohm impedance and you need to maintain that through your entire project to avoid distortions and noise being introduced into the output.
There are ICs made especially for switching video signals.
Use your favourite search engine to search for "video crosspoint switch ic"
You can combine several of them to make up the matrix size, but as DaveX pointed out, you do need 576 switches in total
Video is very broad nowadays. Is it analogue or digital? I did a quick search and found Video and KVM switching; I'm sure that there are others. They have relays for digital video: LEN VIDEO RELAYS - SD, HD, 3G, 4K/12G SDI.
I know that there are companies that sell complete units (probably not 24x24) that can be controlled over e.g. serial or a network; for me that would take a lot of the HF pain away 
Long ago I have written control software to implement redundancy switching with those but I can't remember the manufacturer.
That is correct. I cannot wrap my head around how I want to go about it though. If I use relays, I would need to get tiny signal relays and most likely I/O expanders!!! Using ICs would be the best bet but I don't know of any that switch analog video signals. I ill have to continue to dig.
Analog video signals. From CCTV cameras and monitors. I tried searching the other day for the type of switcher I needed and could only find loads of inputs but only one or two outputs. I can mentally break down all the switching that would need to happen up to around 100-150 relays, but after that I get a bit lost in my brain LOL. I could try to draw a massive schematic of it. But as I stated above I would like to not have to use relays. Space is no issue as I have set aside a few 1U and 2U rack mount chassis for this project. I may have to break out all the inputs and outputs into a separate panel but that is fine. That is cool that you developed software for that gear!! That sounds like it was a fun project 
I'm honestly not sure what the Ohms rating is on the cameras or the monitors. I do know when I hook a camera to a monitor it works pretty well. How can I go about knowing this information so when I do go to build the final unit I can do it correctly as this is for a client of mine. No real deadline as of yet, thankfully LOL.
That is why I was asking what I can use to go between the inputs, arduino, and outputs. I'm well aware of the limitations of the arduino boards. This is electronics engineering not magical voodoo 
I feel like this Max4456 chip would work. All I would need is 3 of them? If I understood the datasheet correctly anyway. All I would need to do is get a library going to control the chip then get the selector sketch banged out. Am I correct in saying this?
Three will only get you a 8x24 fanout. I think you'd need at least 24*24/64=9 of the 8x8 switches. Maybe you can think of it at turning connections off, and you need three behind each set of 8 outputs to connect to the 24 inputs. If a,b,c are three sets of 8 inputs, and A,B,C are three sets of 8 outputs, you need one of the 8x8 crosspoint switches between aA, aB, aC, bA, bB, bC, cA, cB, cC.
OOOOOooooooo. Thank you for bringing that to my attention! I see now. Okay, one more question, according to the datasheet, there needs to be op-amps at the outputs of the chip. I wouldn't need them between chips, correct? Just on the 24 outputs?
I really haven't done this, so take any of my advice as just some guy on the internet, but If #12 is right, none of the switches feed into each other, they are all parallel. The op-amp circuit for each output would need to collect and add the contribution from the three different switches. That is, you would need 8 opamps to combine the aA+bA+cA into the A set of outputs.
I did, about 50 years ago. I thought analogue video was dead.
You will have many problems doing this. DC offset and cross-talk are some of them.
I would test it out with a 74HC4067 breakout board, before falling too deep into the rabbit hole.
Leo..
The coaxial cable should have markings which you can research. Video was exclusively 75 Ohm impedance, but the equipment document should tell. Most radio equipment is designed for 52 Ohms. Really high bandwidth RADAR display is 95 Ohms.
The name finally came to me: Look for "crossbar switches".
And even more info: Look at ICs for the application.
Okay, so in theory, I could hook up a bunch of port expander chips and hook them to the 576 relay matrix and this should work. Not saying that this is what I want to do, but just spitballing here. In doing so I would be able to "bit bang" each and every output individually? Also there would be very little loss on the relay solution depending on the quality of the relays selected? May be a bit power hungry considering 24 coils would be on at all times though. I'd prefer to go with a more solid state solution, so I'm thinking of it in simple terms right now in my head.
The relays would be arranged in a 24 x 24 matrix (just like a giant keypad) then the coils would be controlled from drivers that are controlled by the port expanders hooked to a mega board (all for prototype of course). If I am correct, then I just need to find the correct chip/chips that closely match this theory to do this for me that can be controlled with some lines of code.
Crossbar switches is what gave me the idea in post # 17. Thank you for that. Didn't realize until you said something that it is just a matrix at the end of the day LOL.
Analog video is still very much alive in CCTV cameras and gear. A lot of business and resorts still use this tech to this day. I will look into the chips you have suggested. The one that I posted is around 21 bones a piece. And to buy nine of them just for an experiment that may not work, or maybe blow up a chip or two in the process is not really my cup of tea right now considering the budget is pretty low right now until I ask for more budget. At that point the project may be bined because of the cost.
Thank you, I'll have a look at the gear tomorrow as it is at my shop and I am home right now