This term is for the most part used by online poker players. When players ask “What do you think of my line?”, they are looking to get feedback on how they played a hand, and if they could have taken a different “betting line”.
A simple betting line is to bet, bet, bet. You will see this a lot on betonline poker. As the preflop aggressor, this means you would bet the flop, bet again on the turn, and bet on the river. This is usually the best way to extract value when you’re confident of having the best hand, and lots of worse hands can call you.
When calling out of position preflop and flopping a strong hand like a set or top two pair, a common betting line is to check/call flop, and check/raise turn, since taking the betting lead in the hand allows your opponent to save a bet or two when they would have bet when checked to.