We used to breed parrots and we often had to rear eggs which, for one reason or another, the parents could not or would not take care of.
You need to keep them at something like 103 degrees (Fahrenheit). Remarkably, almost all birds require the same temperature. And you need to turn the eggs over about once a day--roll them, don't turn them end-over-end or you'll kill them.
The incubation period varies by species. For large parrots it's a month, for smaller birds it's much shorter.
Raising pet birds is a big business so there are lots of companies that make incubators. The fancy ones will even turn the eggs for you. If you have duck eggs, which need to cool down once a day to simulate their mother going off looking for food, the high-end incubators will even do that.
But that is only the beginning. The real work starts when the eggs hatch. Chickens are atypical birds that are born with feathers and the ability to stand up, see and eat. Most birds are born without feathers, blind and helpless. You have to continue to keep them around 103 degrees (don't believe me on that, it's been a long time and I could be off by two or three degrees which would be enough to kill all your birds). Now you have to get a brooder, which is like an incubator for hatchlings, although some people manage to do it with a heating pad. You can't let the babies lie flat or they won't develop their muscles. You have to put them in a little padded Tupperware that forces them to sit up or stand up.
And EVERY TWO HOURS AROUND THE CLOCK, DAY, NIGHT AND WEEKENDS, you have to feed them.
You can buy baby bird mix (that's good for most species) and just dissolve it in hot water, that's the easy part. Then you have to pick up the hatchling very gently and stick a syringe in his mouth, simulating what the momma bird does, and slowly fill his stomach with food. You have to let him swallow. Not too fast or it will overflow and go down his windpipe and drown him. He may fuss and spill it. You have to judge when he's had enough.
This will go on for two weeks for small birds and a month for larger birds, although the feeding cycle becomes less frequent and you can start sleeping the whole night. They will slowly grow out their feathers, open their eyes, and develop enough muscle tone to stand up. Then finally they will develop the muscles in their mouths so they can feed themselves.
At this point you have to know what species your bird is and give him the right food. He might eat seeds, fruit, nectar, nuts, oatmeal, bread, meat (he may prefer bugs but he'll eat cooked ground meat) or something more exotic. Most of the birds you find in your garden are seed eaters, so statistically that's probably what you've got.
Then you have to wait for him to be old enough to fly. This can take a couple of months for a parrot, or a few weeks for a songbird.
You may think you're finished now, but keep reading.
What you now have is an IMPRINTED bird. You fed him when he was a baby and his eyes just opened, so his instincts kicked in and identified you as his momma. Now he thinks he's whatever species you are. I presume you're Homo sapiens so he thinks he's one of us. He's not going to fly out the window and hang out with the other robins or sparrows or mockingbirds, he's going to go looking for people and hang out with them. If you have a dog or cat in the house he will have no fear of dogs and cats so he'll hang out with them too.
He will not be able to live on his own in the wild. You have raised him as a pet and you have to keep him.
This is not 100% guaranteed, some birds and some individuals have stronger instincts than others. In some cases, imprinted birds can be successfully reintroduced to the wild. But there's a good chance that he will want to stay with you.
We've done that with orphaned wild birds. It was a lot of fun. But it's a tremendous responsibility and it changes your life.