WHY and HOW do routers break broadcast domains? (and switches don't, unless they employ VLANs etc).
The answer is actually a simple one, and it boils down to protocol implementation and the netoworking function that routers/switches are supposed to perform.
I've taken the liberty f including this image from TCP/IP illustrated, vol.1.
As shown above, end systems (hosts) implement the whole protocol stack, up to the application layer, while switches implement up to (and including) layer2 -datalink, while routers up to (and including) layer3.
Routers and switches as abstractions
Now, someone might point out that there are so-called layer-3 switches, for example.
The answer to this is that 'router' and 'switch' are best thought of as abstractions rather than specific networking equipment.
What I mean by that is that a router is identified by the function it performs - routing and forwarding at layer3 - rather than the physical device that normally represents a router.
To elaborate on this (or otherwise I might only muddy the water), a router running OpenWrt, for example, is quite obviously not limited in its protocol implementation to layer3, as it can run most of what a Linux end system typicallly can.
Additionally, all routers will typically have a Web Interface, which means they run a web server, which entails the transport and the application layers. Switches will sometimes have a Web UI too.
But this is misleading. Although Switches and Routers might implement the transport and application layers, they have nothing to do with their roles : switching and routing, respectively. These are just for management purposes and whatnot - this is what I mean by looking at switches and routers as functional abstractions. An end system can do IP forwarding and routing - and in that case, even though it IS an end system, it ACTS like a router - it performs the FUNCTION of a router.
I made a point of talking about this because you seemed to have the right idea ('act as gateways' etc, but hadn't fully grasped it yet).
A router that acts as a bridge, is, for all intents and purposes in this context, not a router, but a bridge, since that's the function it performs. Since only a 'router' (an actual router, or a device acting as one - again, it could just as well be an end system implementing routing functionality) breaks a broadcast domain and you posited that they were acting as bridges instead, then no, they wouldn't break a broadcast domain.
And how does a router 'break' a broadcast domain in the first place?
Let's say we have a router with two interfaces, addressed at layer 3 as 10.10.10.0/24 and 192.168.1.0/24, with 4 hosts connected to it via wired Ethernet. Host A is 10.10.10.10.
Let's take the scenario where host A sends out a broadcast packet (carrying ARP or whatever else):
hostA builds up the packet, and ultimately encapsulates it into a datalink frame that gets pushed out by the NIC, addressed all Fs (ethernet broadcast)
the frame reaches the router, the frame is a broadcast frame at the datalink layer - all Fs. What happens next is crucial. If this were a switch, the frame, being determined as a broadcast, would be forwarded out all interfaces except the one it originated from. But since this is a router, the frame gets decapsulated and passed up to the network layer. How are IP broadcasts addressed? It would be either 10.10.10.255 or 255.255.255.255 (referring to the current network), which will be determined by the source address - 10.10.10.10/24. This means that the router will determine that the broadcast is meant for the 10.10.10.0/24 network, and that's where it'll forward the packet, sending it out all the interfaces associated with this network.
What this means is that even though two devices connected to the same router are reachable at the datalink layer, and are part of the same broadcast domain at the datalink layer, they might not be so at the network layer. A router being a router will always decapsulate the packet and look at its header at layer 3, which might indicate that the two aforementioned devices are NOT in fact part of the same network, and so the broadcast is limited.
This is how routers 'break' broadcast domains.