A handicap gives one side a head start to even out a mismatch. Quarter lines split the stake across two neighbouring handicaps.
In a handicap market one team is given a goal or points advantage before the match begins. The bet is settled on the result after that adjustment. Asian handicap, which originated in Asia and is described in Wikipedia's article on the subject, removes the draw from football betting by using half and quarter lines.
At -1, the favourite must win by two or more for the bet to win. A win by exactly one returns the stake. That refund is the difference between a whole line and a half line.
At -0.5 there can be no refund. The favourite wins the match and the bet wins, or it does not and the bet loses. Half lines are why you often see handicaps of 0.5 in points-based sports as well: they rule out a tie.
A quarter line divides the stake equally between the two nearest lines.
For a worked treatment with basketball examples, see this Traditional Chinese guide to how handicap bets are settled and why NBA lines so often carry half a point (運彩讓分是什麼).