Correction!
The whole point of NAT is to be a mere temporary solution to the infamous IPv4 problem where the number of available/public IP addresses is slowly dwindling. IPv6 seems to solve these problems. NAT achieves this by translating the source/destination IP addresses and ports and makes it seem like they all come from a single IP address, but the sub-network is inaccessable to the outside world. This in fact allows multiple computers to use the service and have access to the internet.

Unfortunately NAT also has some undesired side effects - it breaks stuff, prevents many protocols from working properly. Normally there are some workarounds for these problems i.e. port forwarding, but even that has certain limitations. For the address translation to work, the firewall needs to classify each packet passing through the router and assign it to a specific connection. This is normally true for TCP connections where a connection can be in one of the several states i.e. new, related, established, etc. Firewalls that use this kind of connection tracking are called stateful firewalls. A very popular stateful firewall is linux iptables firewall, which lives in many routers today.