Rotating the Part,

If you want the part to rotate according to the mouse in someway, this worked for me,

-- LocalScript in StarterPack
local plr = game.Players.LocalPlayer
local mouse = plr:GetMouse()
local part = game.Workspace:WaitForChild("RotatePart")

mouse.Move:connect(function()
    part.Rotation = mouse.Hit.p-- makes the rotation the mouses position
end)

the above script will NOT make the part move, only rotate.

However,

There is a problem with the above, and that's that when you point your mouse at the sky, or distant objects, it makes the part rotate differently. To fix this, I used the following script,

-- LocalScript in StarterPack
local plr = game.Players.LocalPlayer
local mouse = plr:GetMouse()
local part = game.Workspace:WaitForChild("RotatePart")

mouse.Move:connect(function()
    local x = mouse.X
    local y = mouse.Y
    local z = (x+y)/2
    part.Rotation = Vector3.new(x,y,z)
end)

The above worked much better that the above it would seem.

Here's a link that demonstrates this working.

There's an error with the above script as well though, and that's that it does not work with first person.

Moving the Part,

If you however wanted the part to follow the mouse position, you simply get the mouse, and then set the part as the mouses position. Like so,

-- LocalScript in StarterPack
local plr = game.Players.LocalPlayer
local mouse = plr:GetMouse()
local part = game.Workspace:WaitForChild("RotatePart")

mouse.Move:connect(function()
    part.CFrame = mouse.Hit-- makes the rotation the mouses position
end)

However, the above has a problem with the part flying in your face. To fix this, check if the mouses target is the part. Like so,

-- LocalScript in StarterPack
local plr = game.Players.LocalPlayer
local mouse = plr:GetMouse()
local part = game.Workspace:WaitForChild("RotatePart")

mouse.Move:connect(function()
    if mouse.Target ~= part then
        part.CFrame = mouse.Hit
    end
end)

I how that helped.

Good Luck!