The Service engine soon light can come on for literally hundreds of reasons, but only flashes for a specific one with multiple possible causes.
The Engine Control Computer (ECM) runs tests on all the systems that keep the vehicle running within the designed emissions. Anything-sensor, wiring, valve, vacuum hose- that fails the tests that the ECM constantly runs can set the light, and a number code assigned to that particular system is stored in the memory.
A flashing light is a whole different path. Any time the light flashes, there is an engine misfire with the possibility of damaging the catalytic converter, the whole reason for the computer system and a very expensive item.
So, to summarize, the SES light is probably on and the code will probably be for a misfire that gets bad enough that the computer thinks the converter is being damaged.
A misfire that the computer thinks can damage the converter, and actually IS damaging the converter, are also up for debate. For example, a CLOGGED injector will cause a severe misfire that the ECM thinks can damage the converter., yet won't. However, a LEAKING injector can also cause the exact same symptoms and cause severe damage to the "cat".
A misfire can also be caused by a bad spark plug, coil, injector, wiring to any of the previous items, with about half of them causing damage and the other half mirroring symptoms that can damage the cat.
So, the answer to your last set of questions is, yes, there can be more damage done by continuing to drive it, and depending on your mechanical ability, the part causing the misfire can probably be replaced by you, but identifying the culprit by replacing parts without a proper diagnosis can be more expensive than having the diagnostics performed due to the plethora of parts that may be causing the problem.