Hallelujah is actually two Hebrew words.
הַ֥לְלוּ/hallelú,
the command “All of you, praise!” (KJV “praise ye”), and
יָ֨הּ/Yah (KJV “Jah”),
short for
יְ֭הוָה/YHWH,
“Jehovah, the LORD.” When we say hallelujah, or its Greek variant
ἀλληλούϊα/allilúia (Latin and KJV “Alleluia”), we’re literally saying, “Praise the LORD,” which is why many bibles translate it that way.
There are certain Jews who insist the -jah ending of the word absolutely does not refer to YHWH. That’s because they consider God’s name far too holy to say aloud. (Certainly too holy to abbreviate with some nickname like Yah!) But they wanna say hallelujah, and don’t wanna replace it with “hallelu-Adonai” or “hallelu-haShem” or one of their other euphemisms they use, like the Christian substitution “the LORD.” So they claim Jah means something else, like “yea!” Which is kinda ridiculous, considering all the Hebrew personal names which deliberately end in -iah or -jah, such as Elijah, Isaiah, Jeremiah, and Nehemiah. These names are deliberate references to YHWH; their parents wanted the LORD’s name to be part of their names, and remind them who their God is. Most Jews recognize Jah is totally an abbreviation for YHWH—and since it’s not the whole holy name, it’s okay to say it. So they’ll say hallelujah with no hesitation.
My mom once participated in a prayer ministry in Israel. At one point, when they worshiped together, someone got the clever idea to sing a popular worship song together. One that’d been translated into dozens of languages, so each of them could sing it in their native tongue and it’d harmonize, despite the cacophony of different languages. But when they call came to the word hallelujah in the song… no surprise, they all sang “hallelujah” together. It’s the one word we all have in common. It’s probably more universal than the word “okay.”
To pagans, hallelujah is an exclamation of joy. In the Leonard Cohen song (assuming you aren’t more familiar with the version Cloverton rewrote for Christmas) it’s a euphemism for disappointing lust. Some of the pagan stuff has leaked into Christianity, with the result being people who shout “Hallelujah!” at stuff we probably shouldn’t praise God for. But most Christians correctly understand it means “Praise the LORD,” and that’s why we say it: We’re praising God. We’re encouraging and provoking others to praise God. It is phrased as a Hebrew command after all.
