How does damage to the fallopian tubes affect fertility, exactly?

My husband and I are trying to have a baby and not having a lot of luck so far. I am going for surgery to treat endometriosis in a couple of weeks and my doctor is going to check my tubes.

I was told that shooting the dye through is often enough to open them up if they need it, but there is the possibility that they will be blocked. I don’t want to get ahead of myself, but how damaged is too damaged to be repaired? If the tubes are not opened by the procedure does that mean it is beyond treatment?
Anything else related to the topic is welcome.

Thanks.