How do I get a part to drop on a location where my mouse is pointing at?

Whenever you want to know where the mouse is pointing, just use Mouse.Hit.p

Also, don't use deprecated events like mouse.KeyDown. Those shouldn't even still exist, they're just around for legacy compatibility.

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

repeat wait() until player.Character.Humanoid

local humanoid = player.Character.Humanoid
local mouse = player:GetMouse()


local anim = Instance.new("Animation")
anim.AnimationId = "http://www.roblox.com/asset/?id=333356213"


local function createNewPart(Position)
	-- Creates new part @Position
	local newPart = Instance.new("Part", game.Workspace)
	newPart.CanCollide = false
	newPart.BottomSurface= "Smooth"
	newPart.TopSurface= "Smooth"
	newPart.Material = "Slate"
	newPart.Position = Position
	newPart.Shape = "Ball"
	newPart.Size = Vector3.new(4,4,4)
	newPart.BrickColor = BrickColor.new("Dark stone grey")
	newPart.Transparency = 0
end

local function onKeyPress(actionName, userInputState, inputObject) -- The new way we are detecting input
	if actionName == "V" then
		if userInputState == Enum.UserInputState.Begin then
			-- When they hit v, even if they didn't let go
			local newPart = createNewPart(mouse.Hit.p)

			humanoid:LoadAnimation(anim):Play() -- Play Animation

			spawn(function() -- Run on a separate thread
			game:GetService("Chat"):Chat(game.Players.LocalPlayer.Character.Head,"text","Red")
			wait(1)
			game:GetService("Chat"):Chat(game.Players.LocalPlayer.Character.Head,"text","Red")
			wait(1)
			game:GetService("Chat"):Chat(game.Players.LocalPlayer.Character.Head,"text","Red")
			wait(8)
			newPart:Destroy() -- Still 10 seconds before this gets destroyed
			end)
			
		end
	end
end

ContextActionService:BindActionToInputTypes("V", onKeyPress, false, Enum.KeyCode.V) -- Whenever a Player hits V, connect