From your source/link: "The ULN2003A is capable of driving 500 mA into a single channel, up to 250 mA per channel with four channels active, and up to 150 mA with all seven channels active."
It means that each individual channel can handle a constant 500mA current on its own. The 600mA peak current rating means that an individual channel can survive occasional spikes up to that level (you're not to operate it constantly at that current!).
However when the other channels (there's 7 of them in all) are also active, the entire device is capable of handling a total of 1050mA constant current. Assuming all channels are with equal currents, that's the 150mA rating per channel being mentioned. This total device limit is due to having the channels share the same heatsink (package).
Example 1: If first 2 channels are handling 275mA each and channel 3 handles 300mA, channels 4-7 can only have 200mA total (or 50mA in each if equally split).
Example 2: Channels 1-6 have 400mA in total. Although device still has 650mA spare capacity, channel 7 can only pass 500mA due to the max limit on individual channels.