Depending on what you're going to use for the game, you'll need at least the following:
Some kind of input devices -
Typically a trackball
Some buttons
Optionally coin slots
Something to process the inputs, and run GoldenTee Option 1:
If its a real GoldenTee, this will be a rather crappy computer, an ancient cdrom drive, and a hardware key.
I can take a picture of an actual machine this week, as I need to repair one

Option 2:
If its not a "real" GoldenTee, then you'll need to have a computer of some sort running either the PC version of GoldenTee, or Mame running GoldenTee. Note that the current Mame doesn't do newer versions of GoldenTee.
You can take your computer out of its case and mount the motherboard on its own somewhere in your cabinet.
You'll need to put the PSU in the case also to power the motherboard, and any other devices connected to it - eg hard drives etc.
OutputsYou'll need to have some kind of feedback.
This may include sound, and video. Hopefully both

So plan on either building in some speakers, and a small amp, and a video device, or video output, so that you can connect a video device.
Basically an arcade machine is either a dedicated unit or a computer which has joysticks and buttons for input.
Theres not much magic involved.
If you're basing off of existing hardware, then you might need to cable things to standards - eg Jamma (which is how most arcade machines are connected), but essentially its something like the above.
What we do is integrate all the components, package it nicely, and then boast about how cool our machine is

Suggest read the Wiki for starters, then read more of the forum, and then think about what you need to ask about next.
http://wiki.arcadecontrols.com/wiki/BasicsGood luck!