If you disable the DHCP server in your router, you will need to manually configure all the IP addresses on your private IP network; i.e. at least the IP address on the router LAN port and on each of your computers.
If you enable the DHCP server in your router (which is the better option for most people), then all computers and devices connected on your private IP network can get their IP addresses automatically from your router via DHCP.
Your router may be set to receive its public IP address from your ISP automatically via DHCP. In this case the router is being a DHCP client on the WAN side. In some cases you may instead need to use manual IP configuration using information supplied by your ISP - this varies from ISP to ISP but I think the automatic configuration via DHCP is much more common these days.
Using DHCP for private and using DHCP for public IP addresses are really two independent decisions, so be careful to not confuse one with the other.
In most common scenarios, it's much easier to use your internet router for automatically configuring your private IP addresses via DHCP than it is to do it manually. Most routers even allow you to assign static IP addresses to certain clients (e.g. servers, network storage and network printers).
NAT (Network Address Translation) means that for everyone else on the internet, all your devices looks like just a single device with the single IP address attached to your router's WAN port.