The DNS name resolution process involves a DNS server checking its local cache, hosts file, and forwarding the request to higher-level DNS servers if the address is not found. As a last resort, the root hints file is used to forward the request to a root DNS server, which will then direct the request to a top-level domain server that can provide the IP address. DNS translates hostnames to IP addresses through a hierarchical system of root, top-level, and authoritative DNS servers.