Bread is one of the most common things people offer birds — both pet birds and wild ducks at the park — but it is far from an ideal food, and for wild birds it can genuinely cause harm.
The short answer
Bread is best avoided. A tiny piece of plain, unsalted bread will not poison a pet bird, but bread is essentially empty calories — it fills a bird up without giving it the nutrition it needs, and shop bread is often salty, which is bad for a bird's tiny kidneys.
Why bread is a poor food for pet birds
Birds thrive on a base of pellets or a good seed mix plus fresh vegetables. Bread offers little of value, and because birds are small, a few beakfuls of bread can crowd out the food they actually need. If you offer it at all, make it a rare, crumb-sized piece of plain wholegrain bread — never a regular food.
Bread and wild birds (ducks and garden birds)
This is where bread does real damage. Feeding bread to ducks and geese fills them up so they stop foraging for nutritious food, which can lead to malnutrition and deformities like "angel wing" in young waterfowl. Leftover bread also rots in the water, spreads disease and attracts rats. Far better wild-bird foods include sweetcorn, defrosted peas, oats, chopped lettuce, seeds and halved grapes.