If you have a sick , injured or orphaned pigeon please do not feed it immediately, it may be suffering from shock and/or dehydration.
It is important to give it a brief examination to establish any first aid that may be required. First check it for bleeding and stem any bleeding by direct pressure with a clean cloth. If it is bleeding from the beak or a claw you can dip the beak or claw in cornflour to help stop the bleeding.
In hot weather it is particularly important to check it for maggots/fly strike. . The unhatched maggots will look like grains of white rice and will be found in the area of any wound, orifices such as the vent and sometimes at the base of feathers
Check the front of its breast for damage which could indicate a ruptured crop.This is common in pigeons, it is repairable but will need a vet to suture both layers of the crop .
Check under its wings for wounds which could indicate it had been caught by a cat (or dog). Cats carry the pasteurella bacteria in their saliva and a pigeon can die of pasteurella septicemia within 24 hours, so it is important that it is treated with antibiotics (preferably one that combines amoxicillin with clavulanic acid such as Synulox, Clavamox or Clavaseptin) as early as possible.
If it feels cold it is important to warm it up on a hot water bottle wrapped in a towel (so it is warm but not hot) or on a heat pad set low, or under a 40 watt angled lamp for about an hour. If the pigeon is wet and hypothermic you it is best to use a hair dryer to dry and warm it but make certain that the air reaching the pigeon is not too hot. (Please note that if a bird is suffering from concussion heat could be harmful)
After the pigeon has been warmed (that will take about an hour on a heat pad) mix 1 pint warm water with 1/2 tablespoon of glucose, or honey or sugar and half a teaspoon of salt. Dip the bird's beak in the warm water to encourage it to drink, you can also dribble it at the edge of the beak, but don't try squirting the water into the birds mouth as it might aspirate it and die or develop pneumonia.
Do not try to feed the bird yet. It might not be well enough to digest the food, which could kill it. The glucose/honey/sugar will provide the energy it needs for the time being.
You should also check the inside of its mouth which should be clean and pink. Cheesy growths in the mouth could indicate canker.
Pop a clean white kitchen towel under the bird so that you can monitor its poops. Ideally these should be plump and soft but not runny, of a khaki colour, topped by a cap of white.
There are many variations of poop that indicate different health problems: blood in the poops could be a sign of parasites (worms or coccidia) a bacterial infection (salmonellosis, e-coli) or a protozoal infection (Hexamitiasis). A solid worm shaped poop in a splash of water is typical of the effect of Paramyxovirus on the kidneys.
If the pigeon has diarrhoea it will need to drink sufficient water to avoid dehydration. Smelly dirrhoea is typical of a bacterial infection such as paratyphoid, which would need to be treated with an antibiotic like Baytril.
Isolate it from any other birds that you have for at least 2 weeks, birds can be infected with pigeon paramyxovirus and shed the virus before showing symptoms.