Is it possible to detect what part is touching the other part?

Hello!

To answer your question in brevity, yes it is possible. The Touched event has a parameter (part in the snippet below) that refers to the other part touching the part in question.

BasePart.Touched:Connect(function(part)
    print(part.Name)
end)

You can read more about the Touched event here and see a working example here.

Another way to do it is by using the FindFirstChild() method (NOT a function; functions do not run under a class or object):

part = script.Parent
if part:FindFirstChild("Part") ~= nil then
    print("vaild Part")
end

Both the Touched event and the FindFirstChild() method contribute to something touching another thing.

Other things that can be touched/clicked

ClickDetectors have the event MouseClick to detect whenever the Parent of the ClickDetector is clicked. It is used as so:

part = script.Parent
part.ClickDetector.MouseClick:connect(function()
    print("Part has been clicked")
end)

In Guis, this event has two names: MouseButton1Down and MouseButton1Click. They both do the same thing whenever a Text or ImageButton is clicked, so it doesn't matter:

gui = script.Parent
--[[
1st example, which uses MouseButton1Down
This event is used to detect whenever a player left clicks on a Gui
--]]
gui.MouseButton1Down:connect(function()
    print("Valid Mouse Input: left Click")
end)
--[[
2nd example, this one uses MouseButton1Click
Again, they do the same thing, so don't be confused
--]]
gui.MouseButton1Click:connect(function()
    print("Gui has been clicked")
end)    

Now, there is something in Roblox called raycasting. Raycasting is when a ray touches a Part. I can't explain it without being complicated, so you can look for it on this site, the Wiki. or YouTube.