In my router I can set 1 local dns addres and 3 static dns addresses.
What are the differences?
Why do I get only 1 choice for local dns address but 3 choices for static dns addresses?
These two fields, although they're maybe a bit confusingly named, refer to very different things. Here's how I understand it:
The Local DNS field tells your router where client computers should find the DNS server on your network. When computers DHCP (ask the router what settings they should use), they will be told to use this IP address as their DNS server. You will usually just leave this blank, which for most routers that provide this option will result in the router itself acting is a local DNS server (to resolve local machines and serve as a proxy to query public internet DNS servers).
But wait, how does your router know what public internet DNS server to use? Your router is also a DHCP client, and asks your ISP what settings it should use when it turns on. This is a dynamic configuration, though, the static DNS fields allow you to set this manually if you want to (so the router won't use the settings your ISP suggests). So this option overrides the DHCP settings (or is required if your router is set not to act as a DHCP client at all, which a very small number of ISPs require).
There are multiple spaces for Static DNS settings because it is common to configure two DNS servers for resolving public domains. This is simply for redundancy, in case one is down. But on a small network like the router is intended to serve, it's extremely unlikely that there would be more than one local DNS server. Particularly since the router usually serves this role.