I was raised as a Roman Catholic from birth. I remember seeing these letters on the altar in church. Much later in life, years after I returned to the Church, I assumed that they stood for the Latin "in hoc signo" which means "in this sign." "In hoc signo vinces." was a revelation to the Roman Emperor Constantine in his conflict with his principal rival, Licinius, for the throne. A natural conclusion, but an incorrect one.
It turns out that IHS is a transliteration of the Greek ΙΗΣ (iota, eta, sigma), which are the first three letters of the holy name of Jesus in Greek (Ιησους, which is pronounced almost like the Spanish "Jesus", except the initial sound is not an "h" but a "y"). Use of this symbol, "IHS", or ΙΗΣ, goes back at least to Medieval times.
In Acts 4:12, we find "Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved." I write this on January 3, the feast of the holy name of Jesus.
Awareness of this expression goes back to my earliest years. My grandfather was a member of the Holy Name Society in our parish.
The name Jesus literally means "God saves." You wouldn't know that, though, without tracing it back to its origin in Hebrew. Before we get to the Hebrew, though, we need a brief stop in Aramaic, the language spoken by the people of Israel in Jesus' time. That name, transliterated into English, is "Yeshua". It should remind you of "Joshua", because it's the same name. We hear the name pronounced in Aramaic in one of my favorite scenes from "The Passion of the Christ." See this segment.
In Hebrew, the name is transliterated as "Yehoshua." (See Joshua.) It's at this point that we can detect the components of the name. (From the Wikipedia article, "The English name 'Joshua' is a rendering of the Hebrew language 'Yehoshua', meaning 'Yahweh is salvation'. The vocalization of the second name component may be read as Hoshea—the name used in the Torah before Moses added the divine name (Numbers 13:16)." [The original name of Joshua was the same as Hosea, the name of one of the minor prophets, which means "salvation". Moses added the "Ye" prefix, indicating the name of Yahweh.). From this origin, we derive the name "God saves." (See Matthew 1:21.)
It's quite appropriate that Moses gave his aid, Hosea, the name Joshua, since Joshua was to lead the people of Israel into the Promised Land. Just so, Jesus leads the children of faith into the promised land of heaven.