Go read the Disclaimer again. I am not a doctor. This is not medical advice. Seriously.
Breastfeeding is not supposed to be excruciatingly painful, but sometimes it is. Plugged ducts feel like hard parts on the breast. When they become warm or hot, red streaked, and you feel achy, run a fever and otherwise feel like you have the flu, they are considered mastitis. Without treatment, a blocked duct may become an abcess, which in the past was always surgically drained and usually the mother was told to wean. Not so, any more. Currently, a lot of doctors prescribe antibiotics for mastitis, and many will tell the mother to wean or pump-and-dump while taking the antibiotics, as a "precautionary", better-safe-than-sorry measure. This is completely ridiculous, because feeding the baby formula is a huge hazard, and babies often receive the same antibiotics in a much larger dose than they will receive through the milk. Prescribing antibiotics is dangerous to the mother, as it puts her at risk for an equally painful yeast infection or thrush. In the future, we will stop prescribing antibiotics for mastitis, even though they work to fix the bacterial infection, because they so often cause an equally significant problem that must then be fixed in a way that can easily be used to fix the mastitis.
I had two personal experiences with mastitis, one on Thanksgiving Weekend, and the other was a Christmas event.
Most books on how to breastfeed devote some amount of space to correcting plugged ducts and mastitis. Here is what I, personally, found to be helpful, based on my holiday experiences.
It's worth trying to nurse on the affected breast or breasts, however, if your baby is not nursing well on those breasts, or unable to drain those breasts, it helps a lot to drain them some other way. If you are already pumping, pumping probably is a good way to do this. I was not pumping when I had my two bouts of mastitis, and I can tell you from personal experience that attempting to figure out even the quite simple Avent Isis while suffering from mastitis does not work well. I was able to hand express, however, although my technique then was quite poor. Learn to manually express breastmilk. It serves many purposes.
Ice or cold applied to the painful part of the breast or breasts between feedings can help a lot to reduce inflammation. A lot. This is so much more effective than taking pills, it bears repetition. Use cold to reduce inflammation.
Whatever analgesic (tylenol, ibuprofen, etc.) you normally take for pain is certainly safe enough to take while nursing.
Applying heat before a feeding can help get the milk flowing. Something as simple as a washcloth soaked with hot water from the tap can make a big difference.