Use CFrame.

There's a whole article on CFrames, and this question has already been asked countless times.

Here's an example:

-- local script (to make things easy as an example)
local plr = game.Players.LocalPlayer
local char = plr.Character or plr.CharacterAdded:Wait()
local pp = char.PrimaryPart
while not pp do -- might not be needed; tbh this looks stupid; but it'll work with R15 and R6
    wait()
    pp = char.PrimaryPart
end

local offset = Vector3.new(5,9,0) -- +5x, +9y, +0z

local part = Instance.new("Part")
part.Anchored = true
part.CanCollide = false
part.Parent = workspace

while true do
    part.CFrame = pp.CFrame*CFrame.new(offset)
    wait()
end

How it works

BlueTaslem has answered how CFrames work, and how this works, but I couldn't find his answer because there's no favorite option on ScriptingHelpers. sigh

But, to put it simply:

part1.CFrame = part1.CFrame*CFrame.new(vector3 to move it relative to part)

Good luck!