How do I know the IP addresses of other computers in my network? Say that I have two computers connected to the same network.
I know that from computer1 and file sharing enabled I can snoop around inside computer2.
But say that there is something interesting on computer2, which you access by typing localhost:9091 in safari. (e.g. transmission's web UI).
To access that I would have to type in the ip_address_of_computer2:9091 on computer1.

But how do I find out computer2's ip address without using computer2?
Method #1

The easiest way would be to access your network router's administration page. It will have information about any other devices on the network, including IP address.

Method #2

If you know the computer's network name you can ping it in the Terminal. It will return the computer's IP address.

Method #3

If you don't know the computer's network name, there's another trick you can do using ping. Find your IP address and your subnet mask. Both should be visible in the Network preference pane of System Preferences. Line up your IP address and your subnet mask, and replace any 0 values in the subnet mask with 255 in the same relative position of the IP address. For example, if you have the following IP address and subnet mask, respectively:

192.168.1.151

255.255.255.0

The 0 is in the last field of the subnet mask, so you replace the last field of the IP address with 255 and ping it: ping 192.168.1.255

You should get a response with the IP address of any device on the network capable of responding to pings.

Method #4

Last trick is to use the terminal command arp -a. This will show the IP and MAC address of all devices on the network that it knows about.

Last two tricks courtesy of this Macworld article.