A stable line is a line that doesn't change. You get one when a line comes up as 7 or 8 among the line values:
They're called "young" because they still have room to grow. They haven't reached their extreme, so they don't turn over. A moving line, by contrast, is "old": it has hit its limit and is transforming.
Its role in a reading
Stable lines are the background of the situation. They build the hexagram and its trigrams, so they shape the whole meaning, but in the most common practice their line text isn't read. The focus goes to the hexagram's own text and to any moving lines.
When all six lines are stable, the situation reads as a still picture: what the hexagram describes is what's there, with no turn in sight. There's no relating hexagram. In that case, take your time with the judgment and the image. Your answer is there.
How often they appear
They're the most common result. With three coins, 7 and 8 each come up 3 times in 8, so three out of four lines are stable. With yarrow stalks it's similar: 7 comes up 5 times in 16 and 8 comes up 7 times in 16.