There are a number of words in Hebrew that have made their way into English. One of them is Hallelujah.
Hallel means “praise.” There is an entire service of praise we sing to praise all the many attributes of God. We sing Hallel on all major festivals, on Rosh Chodesh, and at Chanukah. It includes parts of several psalms (notably Psalms 113-118) and other prayers, and hallelujah in various forms is repeated many times.
The “oo” sound in the middle lets us know that in this case, hallel is actually a verb. Hallelu means “We praise.”
Finally Yah (often transliterated “jah”) is one of the many names of God, possibly a shortened form of the Tetragrammaton, the name of God that Jews do not pronounce. In the Bible, Yah appears in Psalm 68:5 (in a Jewish Bible) and Psalm 68:4 in other Bibles. We also see it as part of names: Elijah means “My God is Yah;” Isaiah means “Yah is salvation;” and Hezekiah means “Strengthened by Yah.”
Thus Hallelujah means “We praise God,” which is exactly how it is used by both Jews and Christians. In pop culture, we most often hear the word used by fundamentalist Christians, but the origins of the word are Jewish and in fact, observant Jews sing or pray psalms every day containing the word.