Thanks for asking this question. First a bit of background.
A modern wireless device like a wifi router is responsible for implementing all 4 layers of TCP/IP stack.
The link, internet, transmission and application are all implemented in the same box.
Of these 4 layers only the link is the actual physical component. Rest all the layers are logical and as such they are provided by the software inside the router.
A router has its own operating system that manages these software services automatically.
DHCP or dynamic host configuration protocol is one such service. It works on the application layer of the stack. When the router turns on, it automatically starts the DHCP server. From that point DHCP leases local IP addresses to each connected device. When a device is no longer connected to the network of that router that IP address is reclaimed.
As you might have guessed Opensource router operating systems give you a lot of freedom in configuring DHCP server. Where as closed source systems are a bit limited in this regard,but still good enough for normal use cases.
By the way, there is no difference between working of DHCP on a wired/wireless router. The protocol is independent of the link layer. Whether it’s a wired network or a wireless it’s all the same as far as DHCP is concerned.
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