There's a couple ways to do this, but the easiest is to assign two network addresses to the interface on Computer One. Since you're on the same physical network, you can do this pretty easily. This is called multi-homing. The only drawback to this is that you can't use DHCP. You'll have to manually assign both addresses on the computer.
Before you begin, you should know the IP address and subnet mask that Computer One will have on the normal network (192.168.2.xxx), the DNS server(s) that that network uses, and the default gateway for that network. You should also know what IP address you want Computer One to use on the POS network (192.168.1.xxx), and the DNS server that network uses (if any). You do not necessarily need a default gateway for this network.
Computer One is a Windows 7 system, so go Start → Control Panel → View network status and tasks → Change adapter settings (listed on the left). Right-click on the active network adapter on Computer One, and choose Properties. The properties dialog box should appear.
Select Internet Protocol Version 4 (TCP/IPv4) and click Properties. Another properties dialog box should appear. On the General tab, change the setting to Use the following IP address then click the Advanced button. The Advanced TCP/IP settings window will open.
On the IP Settings tab, under IP addresses, click Add, and add the IP address and subnet for the normal (192.168.2.xxx) network. Repeat, and add the IP address and subnet for the POS network. Under Default gateways click Add and add the default gateway for the 192.168.2.xxx network. You can also add the default gateway for the POS network, but this should not be necessary. I wouldn't add the POS gateway unless you know you need it for some reason.
Now click on the DNS tab. Click the Add button, and add the DNS server(s) for the normal network and the POS network.
Click OK to close all the open dialog boxes, and you should now be able to ping the POS computer's IP. You should be able to use the POS computer's hostname, but you will probably have more success with the IP address.