In this image we see a house-spider egg sac and its recently hatched spiderlings (who are about three days old and the size of a pepper grain). Although Charlotte isn’t a house spider, she also produces an egg sac containing hundreds of eggs. Photo by Richhoyer99; online via Wikimedia Commons. License: CC BY-SA 3.0
Female spiders lay their eggs inside an egg sac which looks like a cocoon. It depends on the species, but there could be hundreds of eggs. The egg sac, like a spider's web, is made of silken thread.
Some females carry their sacs - in their jaws or on their spinnerets - until the eggs hatch. Other species hide their egg sacs under a rock, attach them to a plant stalk or encase them in the web. When the spiderlings (baby spiders) hatch, they are usually on their own.
Let's examine the egg sacs of a few spiders:
As part of the life cycle of some spider-species, the mother dies after she lays her eggs. When the eggs hatch, the spiderlings will be able to make their way in the world.
Although they do not have wings, the baby spiders will "fly" to different locations by "ballooning" on lines of gossamer silk.