I have a netgear router and I want to host a local website on that wifi network. I want anybody connected from a laptop or mobile phone to be able to type "mylocalwebsite.com" in the web browser and be redirected to the ip of the server where the website is hosted. Currently the only way I can access the website is by manually typing the ip address of the server, but I want people to visit my website by typing a more user friendly name. Can I do this just by using my router or do I need to set up a DNS server? Or is there another way?
May be depending on the model, but for most you could manually point it in the DNS settings on the router. See this KB.
It's worth nothing in all likelihood they could use the hostname of the server in most conventional network environments too. Hostnames are usually prettier than IP addresses?