how do I detect a key pressed by a player (for example the '' E '' key)?

Sir I believe the event that you are using is deprecated (keydown), meaning its no longer recommended to be used. Instead, use UserInputService.

local uis = game:GetService("UserInputService")
local player = game.Players.LocalPlayer

uis.InputBegan:connect(function(key, gameevent) --Second paramater check if input is game related such as typing in chat *I believe*
    if gameevent then return end
    if key.KeyCode == Enum.KeyCode.E then -- checks if key pressed is E
        print('E has been pressed')
    end
end)

Don't use mouse.KeyDown, its deprecated. Instead, use UserInputService Here's a basic example of how to activate a function when the E key is pressed:

local UIS = game:GetService("UserInputService") -- Get UserInputService

UIS.InputBegan:Connect(function(input, gpe) -- input is the InputObject and gpe is gameProcessedEvent, this is to make sure the function doesn't activate if the user is chatting
    if gpe then return end -- If the player is chatting, do nothing
    if input.KeyCode == Enum.KeyCode.E then -- If the KeyCode is E then
        print("Hello World!")
    end
end)

You can also use UserInputService to detect mouse presses

local UIS = game:GetService("UserInputService")

UIS.InputBegan:Connect(function(input)
    if input.UserInputType == Enum.UserInputType.MouseButton1 then
        print("Mouse Click")
    end
end)

You should look at these threads to know what UserInputService can do UserInputService: http://https://developer.roblox.com/en-us/api-reference/class/UserInputService The KeyCode Enum: https://developer.roblox.com/en-us/api-reference/enum/KeyCode Types of userinput: https://developer.roblox.com/en-us/api-reference/enum/UserInputType