What do these forward currents at specified wattages mean?
As others suggest - a LED without a datasheet is a black box.
That said, you might cull some information from a decently specified 3W LED made by Broadcom on its data sheet: Their device: ASM3-SxDx-xxxxH-DS100
Figure 7 below shows voltage and current relationship for a few different colour LEDs. The royal blue or cool white LEDs are somewhat similar to your meager specs. Figure 7 would be pulse-measured so that ambient temperature remains at 25 degrees C.
Figure 12 shows how temperature modifies forward voltage. It is very likely that your LED will need a heat sink to keep cool - since you have no specs, how much cooling you need is unknown. Your LED might do for just foolin' around, or for running much less than the limits shown here.
Most LEDs are best driven with a current source, rather than a voltage source.