Needing to know how either return or nil variables work in my situation?

Now as far as your problem is concerned, you actually don't need any of these solutions. Instead of connecting to a different function depending if 1Up or 1Down is pressed, connect to the same function, but send a small argument that modifies what you're trying to do. Here's a quick example of that:

Server:

--Server
local function animation(argument)
	if argument then
		--play animation
	else
		--stop animation
	end
end

RemoteEvent.OnServerEvent:Connect(animation)

Client:

--Client
--1Up
RemoteEvent:FireServer() --Sent nothing so play the animation

--1Down
RemoteEvent:FireServer(1) --Sent something so stop the animation

Of course you'll need a way to prevent the client from spamming the button, so most scripters do a check on the client first (normal player spammed button), and on the server (exploiters can work around your client check and spam it anyway)

For completeness, I'll also explain returns. Returns simply just return whatever value(s) you specify back to whatever called the function. Extra values returned that aren't set as a variable on the calling end will be completely ignored. If no return value is set, then the default value is nil. Here is a little cheat sheet I just made so you can see some of the different ways return can be used:

local function sample1(argument1, argument2) --argument2 is nil because we only passed one value
	print(argument1, argument2) --prints 0 nil
	return argument1 + 1 --returns 1 because argument1 is 0
end

local variable1, variable2 = sample1(0) --variable2 is nil since we only returned one value
print(variable1,variable2) -- prints 1 nil

And here are a few more examples of the functionality of return:

local num, num1 = 0, 5  --sets num as 0 and num1 as 5

local function sample1(argument1, argument2, argument3)
	if argument1 then -- if argument1 is not nil/false then
		return argument1 + 1
	elseif argument2 then
		return --This returns nil
	elseif argument3 == 5 then
		return sample1(argument3) --This will recall the same function and replace argument1 with argument3, then return that value
	end
	print("This will not print because return stops subsequent code from running")
end

local variable1 = sample1(1)
print(variable1) --prints 2
variable1 = sample1(false, 0) --avoiding the first if check
print(variable1) --prints nil because the elseif returned nil
variable1 = sample1(nil, nil, 5) --Accesses the argument3 check
print(variable1) -- prints 6