How can I Tween a GUi's position when a ClickDetector is clicked?

(Note: Please format your post correctly, and maybe add some more detail about what you want to achieve next time.)

To answer your question:

script.Parent.MouseClick:Connect(function(player) -- MouseClick event returns the player that clicked it.
    if player then
        -- LocalPlayer only works for LocalScripts. Since "player" returns a Player instance, use it to get the ScreenGui.
        player.PlayerGui.ScreenGui.checkin:TweenPosition(UDim2.new(0.483,-200,0.69,-200),'Out','Quint',0.3,true)
    end
end)

ClickDetector.MouseClick event returns a Player object, which is basically the player who clicked it.

Also, server scripts cannot access LocalPlayer, only LocalScripts.