In classes devoted to poetry we often are exposed to poetic or proper meters -- iambs, trochees, and the like. But it seems to rare for people to discuss syllabic meter, which is unfortunate, because most of the poetry to which we are actually exposed in English, like songs, have no consistent proper meter, building instead on syllabic meter. "Twinkle, Twinkle, Little Star" quite literally shifts proper meter from line to line, but has a very consistent syllabic meter. It is this contrast that makes it so catchy, in fact: the syllables for each line are unchanging, so they all sound very similar, but the lines vary in poetic meter so they each sound subtly different. Hymns work on the same principle, and one sign that our ability to appreciate hymns properly has failed is our inability to recognize how modular they are.
While there are exceptions, most hymns have a stable hymn-meter. For instance, "Amazing Grace" has the hymn-meter of 220.127.116.11 (sometimes written 86.86). That is, it has eight syllables in the first line, six syllables in the second, eight in the third, six in the fourth. This is what is known as ballad meter or Common Meter, CM for short. As the names suggest, it is also extraordinarily common meter. Another common hymn-meter is the Long Meter (LM), 18.104.22.168; "Praise God from Whom All Blessings Flow" is an example, as is "O Come, O Come, Emmanuel". Also common is the Short Meter (SM), 22.214.171.124; "And Am I Born to Die" is an example. There are hymns that have a stable hymn-meter of a more uncommon kind; "The Battle Hymn of the Republic" has the meter 126.96.36.199.
A small minority of hymns, like "O Come, All Ye Faithful" and "Silent Night," have irregular hymn-meter. A notable difference between these irregular hymns and the regular ones is that they are heavily tune-dependent: the syllables have to be stretched and shortened to fit the tune, and thus trying to sing "Silent Night" in a different tune is a very tricky thing. This is not so with regular hymn-meters: any lyrics with a given hymn-meter can be sung to any tune that goes with any other lyrics of the same meter. For instance, "Amazing Grace" and the theme song to Gilligan's Island have the same syllabic meter; so you can sing the words of "Amazing Grace" to the tune of the TV theme song, and the words of the theme song to the tune we usually use for "Amazing Grace". "The House of the Rising Sun" is another well-known popular song in CM. And so it goes. Any song with the meter 188.8.131.52 can be sung to the tune of "Twinkle, Twinkle, Little Star". Any hymnal of genuinely good quality will tell you the hymn-meter for the hymns, and have a metrical index linking hymn-meters with tunes that go well with them.
Once upon a time hymns did not come prepackaged the way they do now; lyrics and tunes were detachable. The tune for "Amazing Grace" originally was called "New Britain"; it was a separate thing, and many different songs were sung to it. Then the two began to be locked together, and now people have difficulty thinking of them apart. Thus instead of singing being a handful of tunes to which an endless variety of lyrics could be sung, we have a different tune for every lyric. This makes public singing much, much more difficult. With the old way, you can sing anything together: everyone knows the tunes, and all you have to know from there is the hymn-meter. People who have practice doing this can take a metrical psalms book and, right from the get-go, without any sheet music, sing almost any psalm. Without it, singing becomes less public: those who can, do, and those who can't, stand around and try to look like they are singing, and we start getting the idea that songs are things that people sing at you. Sometimes they are. But song is a standard mode of human life, an expression of human reason; it's something for every voice, not for just the pretty and powerful ones. It's something we can all share in, something that is capable of being part of our shared culture; and we have for a very long time been losing that important idea, which was once so common and powerful that much of the 'soundtrack' of people's lives was built on it, whether they sang hymns, or ballads, or anything else.
Likewise, it meant that anyone could make their own hymn (or, indeed, any song), just as Julia Ward Howe made up her own lyrics to the popular song "John Brown's Body", which became the "Battle Hymn of the Republic". When you can detach hymn-meter and tune, you don't worry so much about things like rhyme: all you need to do is to get the syllables right. It might not be at the level of Julia Ward Howe or Fanny Crosby, but it need not be. So not only did it give a shared culture, something everyone could have in common, it gave a shared culture that was extraordinarily customizable, so each could use that common culture in their own particular, quirky way. There are very few things like that, and in drifting away from the old mix-and-match system we've lost a great deal.