What is the step-by-step process to ping a host on two hosts using a router?

Host1----eth1_router------eth2_router-----Host2

Let's configure eth1 of router as 192.168.10.1 mask of 255.255.255.0
Let's configure eth2 of router as 10.10.10.1 mask 255.255.255.0
Host1 IP is 192.168.10.100 mask 255.255.255.0 default gateway 192.168.10.1
Host2 IP is 10.10.10.100 mask 255.255.255.0 default gateway 10.10.10.1.

Now on the router you may need to add two statements for default routes.
Ip route 192.168.10.0 255.255.255.0 eth1
Ip route 10.10.10.0 255.255.255.0 eth2

Now after doing this you need to do ping 10.10.10.100 on Host 1 or 192.168.10.100 on Host 2; it will go through.

Hope this helps.
-Sudhanshu