How to fix this Mouse.Target error?

The issue is as the error says -- Mouse.Target is nil. How can this be? It's because time passed (from wait()) after your check that it's not nil.

The simple solution is to save the original one in a variable. Since you say it very many times, that makes sense anyways. You should do the same thing with the local player:

local player = game.Players.LocalPlayer

Mouse.Move:connect(function()
	local target = Mouse.Target
	if target then
		if target.Parent:FindFirstChild('Head') and target.Parent:FindFirstChild('Torso') then
			repeat wait() until player.Character
			local Magnitude = (player.Character.Torso.Position - target.Position).magnitude
			if not player.PlayerGui:FindFirstChild('Description') and Magnitude < 25 then
				local Description = game.ReplicatedStorage['Door System'].Graphics['Description']:Clone()
				Description.Information.Text = target.Parent.Name
				Description.Parent = player.PlayerGui
			end
		elseif target.Parent:FindFirstChild('Item') then
			repeat wait() until player.Character
			local Magnitude = (player.Character.Torso.Position - target.Position).magnitude  --Error right here
			if not player.PlayerGui:FindFirstChild('Description') and Magnitude < 15 then         
				local Description = game.ReplicatedStorage['Door System'].Graphics['Description']:Clone()
				Description.Information.Text = target.Parent.Item.Value
				Description.Parent = player.PlayerGui
			end
		else
			if player.PlayerGui:FindFirstChild('Description') then
				player.PlayerGui.Description:remove()
			end
		end
	end
end)

ROBLOX prefers :Remove() over :remove() and it prefers :Destroy() over :Remove().

There's no reason to use ['Description'], you can just say .Description.

You repeat your computation of Magnitude -- you can do it just once before the if instead. It's also a bad name for distance -- magnitude says how big something is -- you should say what if you want your code to make sense!

You should also be careful that the Character actually has a .Torso. You might as well just use an if instead of waiting until it's there.

You also repeat a lot of the code for showing the message. It's probably simpler to just make a message variable and act on its contents after.

local player = game.Players.LocalPlayer

Mouse.Move:connect(function()
	local target = Mouse.Target
	if target then
		if not player.Character or not player.Character:FindFirstChild("Torso") then
			return
		end
		local distance = (player.Character.Torso.Position - target.Position).magnitude
		local message -- What to say
		if target.Parent:FindFirstChild('Head') and target.Parent:FindFirstChild('Torso') and distance < 25 then
			message = target.Parent.Name;
		elseif target.Parent:FindFirstChild('Item') and distance < 15 then
			message = target.Parent.Item.Value;
		end
		if message then
			if not player.PlayerGui:FindFirstChild("Description") then
				local description = game.Replicated["Door System"].Graphics.Description:Clone()
				description.Parent = player.PlayerGui
			end
			description.Information.Text = message
		else
			if player.PlayerGui:FindFirstChild("Description") then
				player.PlayerGui.Description:Destroy()
			end
		end
	end
end)

This is much simpler -- less redundant -- and hopefully easier to work with.