I really don't want to put carbon in my tank. ... How do you know when you need carbon chemicals?
People don't like carbon because it soaks up nutrients/impurities.
I think two different types of carbon are being discussed here, and it's important to not confuse the two.
Carbon, such as CO2, or a chemical form of carbon, such as Seachem Excel, does not soak up nutrients or impurities. It is used by plants as the physical building blocks to make new plant matter when growing (new leaves, stems, etc.).
The chunks of black carbon that is used in filter media is very different. While it does soak up impurities, it does not soak up nutrients. It is generally not needed for most aquariums so it's not used; however, it can be useful for removing tannins and medicines from the tank water. It has nothing to do with plants since it's not mixed with the water, but rather remains in a solid form.
The carbon Django was talking about, I believe, is the carbon plants use for growth. The carbon Zenzu was talking about, I believe, is the carbon used in filter media. The two have nothing to do with each other.
I thought that algae grew and prospered when there are excess resources (light, ferts, NO3).
Algae grows when there is an imbalance
of resources, not necessarily an excess. You can have some things in excess and be okay, such as CO2 and ferts. The one thing you cannot have in excess is light. Whatever amount of light you have, you must match it with CO2 (or an equivalent carbon source), ferts, and plant mass. If any of those things are insufficient in comparison to the light, algae will take advantage of it.