When talking about computers, you'll hear a lot about ports. There are two types of ports, physical hardware ports that act as an interface between two devices, and software ports which are virtual ports that function as logical constructs in a network service.
For the purposes of discussion on port forwarding, we are talking about software ports. These ports exist as part of TCP/IP and UDP protocols and are always associated with an IP address. Ports are unique and identified by a 16-bit unsigned number, which is commonly referred to as the port number. Most port numbers are standardized and regulated by the IANA to ensure consistency.
Port forwarding is the digital process that becomes necessary when you are using a local IP address rather than a public one, and are behind a router or modem. With a public IP address, the router or modem knows how to handle the request, but not with local IP addresses. Port forwarding communicates to the router or modem what device on the LAN is making the request, and the port protocol goes ahead as it normally would with a public IP address.
Port forwarding is also a useful tool for cyber-security, providing an added layer of protection against digital attacks by functionally acting as a firewall, preventing additional inbound requests that are irrelevant to the port protocol.
It can be a little bit of work to set up port forwarding, but your modem or router manufacturer should have instructions on how to do so for your specific model. This article by PC World contains some good general instruction, as well.