What is the explanation for the different lines in the payload/range diagram for the Airbus A320?
There are two different Maximum Take Off Masses (MTOM) listed in the graph: 73.5 T and 78 T. This is only a small selection. The Airbus A320 has many variations due to modifications to fuel tanks, fuselage, engines, wings and landing gear. The following MTOMs have been certified (EASA PDF): 66 T, 67 T, 68 T, 70 T, 71.5 T, 73.5 T, 75.5 T, 77 T and 78 T.

In addition to the variation in MTOM, there are variations in the number of fuel tanks installed. The A320-100 series have either 2 or 3 tanks, while the A320-200 series can have up to 5 tanks. More fuel tanks allow an increase in range at the cost of payload. Even on short flights additional tanks will decrease the maximum payloads due to their structural weight. The brown line in the chart represents an A320 with one Additional Centre Tank (ACT) installed. 

Payload includes passengers as well as cargo. Assuming an average adult weight of 88 kg (EASA PDF) including carry-on luggage, 180 passengers weigh 15840 kg. Depending on the variant of A320 you take, that leaves a couple of tonne margin for other payload such as check-in luggage or fuel to increase the range.

For more on estimating aircraft loading, see this question.