I liked him the minute I saw him! (Para. 1)
Meaning: I liked him as soon as I saw him!
"The minute (that)" is used here as a conjunction to mean "as soon as":
Telephone me the minute he arrives! 他一到，你就给我打电话!
"The moment (that)", "the instant (that)", and "the second (that)" can be used in the same way, for example:
Telephone me the instant (that) you get the results. 你一得到结果，就给我打电话。
I loved you the moment (that) I saw you. 我对你是一见钟情。