o'er is just over, written the old short way. Poets and songwriters drop the v and put an apostrophe in its place so the line fits the beat: o'er the hills, o'er the sea. You will meet it in hymns, folk songs, and older poems. Nobody in a shop says o'er. If you use it in ordinary talk, people will think you are joking or quoting. It can mean “across” or “above”, the same as over — it is not a different word with a different job. Though actually, you may still hear it sung with great feeling, which can make it feel grander than over; the grandeur is the music, not an extra meaning.

（詩、歌詞裡的）over，越過、在…之上