Usually, I try to reduce the word I discuss to its essential root by removing prefixes and suffixes. In the case of "am'kha," the suffix may mean as much as the word itself.
"Am" means people, people in the sense of nation, not people in the sense of a bunch of persons. It is often used as a synonym for "goy," goy in the sense of nation, not goy in the sense of non-Jew. If you saw this word standing alone and without the vowels, you would not know whether to read it "am" or whether to read it "im." And therein lies the clue to the meaning of "am" and its difference from "goy."A goy is a collection of individuals thrown together by ancestry and circumstances. They may share characteristics and even values, but a goy is defined externally. An am is a collection of individuals who have a sense of being with each other – a shared destiny that is as much motivated by internal commitments as by external circumstances.
Can a people be both am and goy? Of course. The nation of Israel is referred to as both interchangeably. And can a nation other than Israel be called an am? Most certainly.
What elevates "am yisrael," the People Israel to a different level is that little suffix, "kha," meaning "Your." And the "You" here is, I hope obviously, God. When we address God and call ourselves "am'kha," we are acknowledging the essential role that God's presence plays in defining who we are – including those who don't have much in the way of faith. Am'kha, read without vowels, can also mean "with You," with God.
As the centuries wore on, am'kha took on other meanings. In Yiddish, "am'kha" comes to mean the common folk, the Jews in the pews, popular culture. And because of the way it sounds when pronounced colloquially, "am-khe," it became a sort of secret password for some Jews during the Holocaust, a way to identify each other with an otherwise innocuous couple of syllables.
It is hard to hear the phrase "am yisrael" without hearing the next word in the popular folk-song: "chai." The People Israel lives. It resonates when we say it, and when we say "am'kha," it reaffirms its truth.