Q:

LAN to LAN with different subnet

Is it possible to have 2 routers to communicate to each other via LAN to LAN (not LAN to WAN) despite both have different subnet?  
Topo:
Router 1 (192.168.1.0/24)
Router 2 (10.0.0.0/24)  
For example PC under Router 1 with IP address 192.168.1.180 will be able to communicate to PC under Router 2 with IP address 10.0.0.30.
Diagram :  

A:

It's definitely possible, in fact it's the basis of how routing (routers) work. Through a variety of ways - static routes configured into a local router, updates via a protocol called RIP (Routing Information Protocol) which is essentially routers sharing their routing info, and if all else fails, Default Gateways (DG). 
DGs are where traffic gets sent when the local router doesn't know where to send it - typically out of your LAN and to the ISP's router. Your PC will have your local router as it's DG, your router will have the ISP's router on it's WAN port as it's DG, and so on.
In the case you are asking about, each router would need to have a static route configured; which, if it were a Linux based router, you would do using commands something like this, with router 1 having a LAN IP Address of 192.168.0.1 and physical Ethernet port connected to the router 2 called 'eth3' and router 2 with IP Addr 10.0.0.1 connect to router 1 on LAN port 'eth0' (arbitrarily chosen)
On router 1:  
ip route add 10.0.0.0/24 via 192.168.0.1 dev eth3

and on router 2:
ip route add 192.168.1.0/24 via 10.0.0.1 dev eth0

Any PC connected to either of these routers trying to talk to the subnet it is not in would need to have the local router* set up as it's DG.
DHCP typically configures this unless you are using static IPs. 
If it had a different DG (and presumably multiple NICs) it would need to have a static route set up locally (e.g. 192.168.1.0/24 via [computers IP 10.0.0.x] dev [computers NIC connected to router 2]. The router would still need the same routes added from above in either case.
*technically the local routers LAN IP address
This site has a good write up on the concepts involved here:
http://linux-ip.net/html/routing-intro.html
If you had RIP version 2 or other more sophisticated inter-router procotols, this routing info could be communicated between the routers without any additional configuration - see 
https://en.wikipedia.org/wiki/Routing_Information_Protocol

