"Another" is basically the combination of the words "an" and "other." If the item that you're speaking about is singular and indefinite, use "another."
Can I have another piece of pizza? (This would mean that there is not a specific piece that you're talking about, but just any piece out of what's left.)
If there are 2 pieces left, and one person takes 1 of them, someone else might say, "I'll have the other one, please." We use "the other" because it is a definite noun (1 certain piece of pizza).
Hope this helps! Please let me know if you need further explanation.