Egg Binding is, to be frank, what happens when an egg gets stuck in a bird. The exact severity of the condition varies bird to bird, (dealing with an eggbound hen is different than dealing with an eggbound parakeet), but it should always be taken seriously.
If the egg binding isn't too bad, often times placing the affected momma bird in a warm place (under a heat lamp, for example), or in a warm bath will help her pass the egg on her own. Sometimes, for larger birds like chickens, the egg can be massaged out. For smaller birds, massaging is a bad idea because that risks breaking the egg inside the bird. The last way to get rid of the egg is via surgery at the vets.
Death by egg binding is pretty much the worst possible death a bird can have. What happens is poop gets blocked in by the egg, so nothing is going out at all. The egg eventually breaks and the pieces of shell tear apart the momma bird's insides, so they die a slow, shit-covered, agonizing death while their insides slowly become their outsides as their cloaca tears open.
Parakeets, especially, are in danger of becoming eggbound as their eggs are basically the size of their heads. Other at-risk birds include canaries, lovebirds, and cockatiels.
Signs of egg binding include: heavy breathing, a swollen or red bottom, straining (as though she is trying to pass a freaking egg the size of her head), shaking and tail waggling, loss of appetite, lethargy, loss of ability to perch (they will be found on the bottom of their cage), inability to poop, and sometimes you can actually see the egg sicking through the vent.