Switch or Router for LAN party?

I was wondering, if I want to play games with lots of my friends, without clogging up my house's router's connection (my grandma will be using it at the time), whether I should buy a network switch with lots of plugs, or buy a router and keep it offline. We would be using Ethernet. Or is a router just a switch with nat/dhcp/whatever?

You would want to get a good switch for sure. It really all depends on if you are planning on everyone connecting to the internet, or if this is an actual LAN party and all traffic will be only transmitted at other machines on the same LAN.

Here is an example setup for a LAN party..

The LAN ports of a typical consumer home gateway router (including "Wi-Fi routers") are actually a switch. That is, they are connected to an Ethernet switch chip (sometimes the switch "chip" is actually a functional cell within the "router's" SoC). So it's like a 4-port switch.

Yes you want a switch.

Routers are the borders/connections between IP subnets. If you had a box that was a real multiport router and in no way a switch, you'd have to set up separate IP subnets (kind of like address ranges) for each port, it would complicate the setup, and add processing overhead and latency. Not something you want for a twitch gaming LAN party.

I assume that you are an end-user capable of configuring your games, sharing data via network and so on, but not necessarily expertised in advanced network configurations.

I would recommend a good switch, but not a professional one (characterized for having "management" capabilities).
I suggest to buy the best you can afford from those who have no management, as management capabilites could add some more difficulties for initial configuration, and some kind of firewalling or defenses against possible LAN attacks (like ARP) who sometimes, in fact, stop all the network communication and requires a rebooting of the switch.

Another detail to remember: for what respect to software configurations, if using a switch, you will not have any DHCP Server. I would then configure the IP/Netmask manually, the best idea if you use a switch.
Windows is supposed to do this if there are no DHCP server on your network, but I use to prefer doing this by hand.

A router would serve DHCP address (IP/netmasks), thus making you easier the IP/Netmask configuration matter. But some routers have a preconfigured limited DHCP Pool (i.e: 50 computers), that could become exhausted (for example in cases of rebooting or computers blocking): again the first configuration issue (small) difficulty with routers.

Furthermore, routers become heat more easily (or more easily than siwtchs), due to they must make more operations, run more threads in its embedded operating system... etc. Even when there is no internet traffic.

The usual method when big load of users is expected in an office is to buy a good router with only a few RJ45 ports, and a good switch with many ports. You are in the case of only needing the second.
Lets remember that party addicts have the (magic) power of stay playing continuously at 100% throttle during hours and even... days (the craziest one do)! Harder for many routers.

On the other side, an advantage of buying a router is the possibility of WiFi: most of them do have it as for today. It is up to you to consider it.
A (rather ridiculous) example: when I played partys, we sometimes sent one only player to another room for making him to play without speaking (requesting help... etc) with others. In this case, deploying one only cable was annoying. WiFi was a valid option.

Exception: if you are an expertised network manager/user, you love dealing with initial configurations ("to make the odd network work!"), and don't mind to troubleshoot devices while your partners are awaiting ("come on, man, I came here to play!") maybe a professional switch will be for you.