>If you have a network printer connected to a router then, I suppose, you can print to this printer from any computer on the network. If two computers try to print at the same time what controls the spooling (since there is no printer server)?

More precisely, typically a standalone network printer is connected to a switch rather than a router. Many consumer-grade routers contain a built-in switch with 4 or more LAN ports, which may be what you're using in this case.

Network printers do contain built-in print servers. They usually run an embedded OS (often Linux-based) which offers a print server/spooler, print processor, and these days usually an embedded web server for administration and configuration. Some higher-end models have lots of RAM and may even have an internal hard drive for spooling large print jobs, holding document templates etc. They are basically computers with a print engine.