If you're looking for childcare for your child under school age at a Kindertagesstätte, or "Kita" for short, then you're in for a tough time. For children under three, there are childcare options at Kinderkrippen (daycare centers or groups). Children from the age of three to six can go to Kindergarten.
Lots of primary schools offer Hort (after-school care) or offene Ganztagsschule (OGS).
Since 2013, every working parent in Germany has had a legal entitlement to a Kita place for their child from the age of one, but it is still hard to find a place close to home. This is especially true in western Germany. There are historical reasons for this. When Germany was divided, it was common in the GDR for women to work and for their children to be in daycare from a very young age. In West Germany, women with young children usually didn't work, and there was no great demand for childcare centers. Today, the situation is different.
If you live in Germany and want to have a child with your partner, then you should find out about childcare options in your area and register for a place even before the child is born.
You pay fees for daycare to the city in which you live if it is a public facility. You can also apply for a spot at a private daycare center. These have their own fees, which are usually higher than public childcare centers.