Can Network Address Translation work without static IP addresses?

Depending on your requirements and network configuration, you can have different types of NAT configured. NAT is...

usually configured on your router or firewall. This allows internal network resources to become available to the public (Internet) and provides Internet access to all of your internal hosts.

The most common type of NAT is NAT Overload (Cisco), also known as NAT with Port Address Translation (PAT). With NAT Overload, all internal IP addresses are masked by your router using your unique public IP address (static or dynamic). This allows hundreds of connections to the Internet by smartly changing the source port of each connection (where required).

Generally NAT is a big topic, and you'll find plenty of information on the IT Knowledge Exchange's NAT question and answer section and www.Firewall.cx, including detailed diagrams and examples to help make the concern clear.