With the greatest respect to Ron Maupin, it's different in Europe.
Over the last 20 years, virtually all fixed line IPv4 Internet connections in Europe use NAT, largely because RIPE (the regional Internet registry) has tried hard to conserve the IPv4 address space by making it difficult to order multiple addresses.
When you order a home Internet connection, you will receive one dynamic public IP address on a router with firewall which typically presents 192.168.1.0/24 (an RFC1918 subnet)1) on its LAN ports and WiFi.
None of the devices on the home network are visible on the Internet for incoming connections (the router does not know which private address maps to its public address), but if you want to host a multiplayer game or view your security webcam remotely, you can configure the router with a static NAT and port entry.
Outgoing connections establish a dynamic NAT entry which maps the private address and port to a public address and port.
Business connections will use static public IP addresses (often more than 1) with one or two routers supplied by the ISP and one or more firewalls supplied by the customer.
The firewall contains static NAT rules which, for example, can redirect incoming SMTP to the mail server and HTTPS to the web server, even for the same IP address.
Most applications work very well behind NAT (FTP being a notable exception, as in active mode the host IP address is sent in an FTP message). Without static IP and static NAT, peer-to-peer applications like Skype need to use a central server to establish the connection (normally, a dynamic NAT entry does not contain the destination address which caused its creation and can (firewall permitting) be used by a third public IP address for a peer-to-peer UDP connection). In general, a central relay server is used for applications such as TeamViewer to get round these limitations.
Apart from conserving IP address space, using NAT protects the devices on LAN from port scanning attacks and permits businesses to use many subnets to create private networks and to use multiple VLANs without involving the ISP.
Another use for NAT is when companies merge and find that they have the same private IP subnets. NAT can be used to enable company A users to access company B's servers without passing by the Internet.