I have an internet connection in my office that goes through a firewall.
It is very slow.
I setup my PC to connect via a tether from my mobile phone. The pc now shows two connections.
How does the computer utilize two connections?
Access to networks that are not directly connected to you are governed by the routing table of your PC.
"Not directly connected" includes every internet based IP address, and you access these via the default route.
When you have two connections that are both allocated an IP address when connected, a default route is also allocated. This means that when you have two connections you will have two default routes. So any packet can take either path. This can lead to issues where different packets from the same session take different routes and so appear to come from different public addresses.
In reality, what usually happens is that the OS will use some smarts and give one of these routes a higher metric, indicating it is a preferred route - so all packets take the same path. This can be based on the connectivity speed for example. So if you have a gigabit LAN connection, and a dial-up 3G connection, then some OSes will prefer the faster connection. Or it may simply be that the connection established second will be preferred.
You can control this by modifying the routing table, and forcing packets to take a particular path by removing one of the default routes, or adding in more specific routes to cause certain traffic flows to use a particular connection.