A concise description of the Windows' "Browser" service provided from a reference site is:
Maintains an updated list of computers on the network and supplies this list to computers designated as browsers. If this service is stopped, this list will not be updated or maintained. If this service is disabled, any services that explicitly depend on it will fail to start.
What is the definition of "browsers" in this quote? What defines a computer to be "designated" as a browser?
The "Browser" service itself does. Any computer that runs it is a "Browser".
One of them acts as master browser and gathers info about network resources and distributes that to the others. If the "Master" stops responding (e.g computer reboots or is turned off) the others will elect a new master among themselves.
Various Windows components like the Network Neighbourhood shown in Explorer and the "net view" command get their information from the Browser service.
The mechanism is well-known to be flaky and unreliable because it is broadcast based, which is a problem in larger networks with many devices. It is quite easy to miss information (broadcast wasn't received or dropped) or for multiple Master browsers to popup because computers thought they had lost the Master system and appointed a new one.
(Broadcast can be done right in larger networks, but the original design of the Browser service was intended for home and SOHO networking with just a couple of computers in a network. It just didn't scale well.)
As of Windows 7 Microsoft has replaced it with other technology (a mDNS variant). The Browser service and its API's still exist for backwards compatibility for older applications that need them, but the information itself is maintained by newer mechanisms.
And in a Active Directory environment the system administrator has usually disabled the service on all computers because Active Directory itself provides the same functionality. (In that case the API's still work, but the backend is provided by Active Directory.)
"Browsing" is a term from the NetBIOS networking suite previously used by Windows and MS-DOS. It is the NetBIOS component which keeps track of computers in the LAN – in other words, it used to provide information for the "Network devices" list.
(It's still present in Windows 10, but only active if you enable SMBv1 support. By default, Windows no longer uses Browsing – it achieves the same through WS-Discovery which has a much simplified process.)
The NetBIOS 'browsing' process is somewhat more complex than e.g. mDNS service discovery (which just directly multicasts a query for the desired service). In NetBIOS, the designated "master browser" is the only machine which continuously keeps track of which other hosts claim which names and publish which services, saving that precious CPU time for everyone else's Intel 80386-powered PCs.
In this protocol, computers designate themselves as the subnet's "master browser" by participating in an election. The computer with highest specs (WinNT, high uptime, etc.) wins the election and provides Browser service to the rest of the LAN, collecting announcements and allowing other hosts to download the entire "browse list" on demand.
(Keep in mind that this was designed for much older networks, back when Ethernet hubs had 'collision' lights and LANs ran five different protocol suites alongside TCP/IP.)
Good resources for more information:
The latter provides these definitions:
browser client: A computer on the network that queries or sends information to a browser server. There are three types of browser clients: workstations, nonbrowser servers, and browser servers. In the context of browsing, nonbrowser servers supply information about themselves to browser servers, and workstations query browser servers for information. Browser servers can behave as nonbrowser servers and as workstations.
browser server: An entity that maintains or could be elected to maintain information about other servers and domains.
local master browser: The browser on a given subnet that was elected to maintain the master copy of information related to a given domain. That is, different domains have different local master browsers on the same subnet.
