It's not the ports on local that are important generally. It's the remote ports.
So when you open a browser and go to a site, you are establishing a connection from a (somewhat) random port on your end, to port 80 on the server end. The server responds back to you on the same connection. Web servers use TCP/IP, so this is what is called and established connection. If you were to go look at netstat -an on the server you connected to during web traffic, that is exactly what you would see:
tcp 0 0 ::ffff:192.168.1.223:22 ::ffff:192.168.1.230:2369 ESTABLISHED
That line says that my local machine has established a connection to my remote machine on port 22. My local machine picked a random outgoing port of 2369 to make this connection.
In this case, this is an ssh connection to my webserver in the basement.
Ports that servers should use for a particular service are listed here, but if you are going to control both ends of the connection, there is nothing stopping you from running a webserver on port 8383 if you wanted to. Just don't expect anyone else to get to it without you telling them about it. (or it being found in a port scan).
If you were running a webserver on your computer, it would open port 80 and listen for connections. Only one service can be LISTENing per IP address, so you couldn't run two web servers at once. Same thing if you then connected to your local webserver. You'd open a random local port and connect to your local port 80 on the same IP.
The opening the random local port is what allows you to have multiple local connections to a known remote port like 80.
There are 65536 ports available so it's unlikely you will ever run out, but many have 'well known' usages and are therefore avoided for your end of the connection. Generally everything above 1023 is fair game though. ( All services which require any kind of priviledge run on ports below 1023 )
This is a TCP/IP connection. TCP/IP has internal language to ensure the reliable delivery of information and does a handshake at the open of every connection to ensure the data can be transmitted.
Another common type of connection would be UDP. UDP does not establish a connection and is therefore a bit faster and has lower latency, but the programs that use it must be able to loose information and still work. It's basically a send off the data and pray protocol. Many online games work this way.