Right, the insulated line is the suction line (after the evaporator and leading to the compressor). It's insulated because it's cold, and the insulation keeps condensation from forming all along that line and dripping all over your house.

It's not what people think intuitively. People think that a cold fludi of some kind is being pumped to the evaporator, but that's not the case. Hot liquid goes to the evaporator under pressure (the high-side, or uninsulated side), and is expanded into the evaporator through a nozzle, causing it to evaporate (hence the name). The vaporization of the refrigerant takes a lot of heat (the heat of vaporization), and it gets this heat from the environment, thus cooling the coil. So the refrigerant leaving the evaporator is cold, but going to the evaporator it's a warm liquid. The cooling state change actually happens right inside the evaporator.