IPv4 addresses are mapped to a domain name by an A (Address) record in DNS (Domain Name System). IPv6 addresses are mapped using AAAA Records. Seems strange? IPv4 addresses are 32bits, IPv6 are 128bits. 128 / 32 = 4, So AAAA / A = 4 as well. Clever trick by whoever designed IPv6's entry in DNS.
DNS is a hierarchical, distributed system across the whole of the internet. Your browser will make a call to a system library, which starts the lookup chain. It'll check its local cache first, usually stored in memory. Next, it seeks to disk and looks at the HOSTS file (/etc/hosts on most UNIXes/Linux, and C:\Windows\System32\drivers\etc\HOSTS on Windows). Then, if it still can't find an answer, it makes a call to your assigned DNS server, per your networking configuration.
THAT DNS server checks its cache, which is much larger. If it has a match, it returns an answer. If it can't find an answer, it turns around and asks an even higher-level DNS server run by the ISP of the ISP. This process continues until you hit the root name servers,
m.root-servers.net. A majority of DNS queries are resolved LONG before they get that far, but sometimes one or two make it up there. But when an answer is found, it's passed right back down that chain to your browser.
But how does your browser know which record type to request? That depends on your system settings. If you've set it to your IPv4 stack is priority, then your browser will only ask for A records in its DNS queries. If IPv6, it'll ask for an AAAA record first, and then fall back to an A record second.
There are other record types, too. MX records define where the mail server for a domain is. NS records define the name servers for a domain. SRV records tell you where a particular service (SSH or Web, for example) are at.
DNS really is the literal phone book of the internet.