NAT serves three main purposes:

Provides a type of firewall by hiding internal IP addresses
Enables a company to use more internal IP addresses. Since they're used internally only, there's no possibility of conflict with IP addresses used by other companies and organizations.
Allows a company to combine multiple ISDN connections into a single Internet connection.
http://www.webopedia.com/TERM/N/NAT.html

Modems in themselves dont have NAT that I know of.Routers have NAT to provide translation but it is not a firewall per se as say software or hardware firewalls which are designed to allow or block traffic to a preconfigfured set of rules.