Next Generation Firewall (NGFW) is partly a marketing term. Wikipedia has a definition (as does Gartner and a host of others). Typically it's understood to mean something more than a simple stateful firewall that only looks at packets up to the TCP session level.
Cisco ASA has had add-on features for years like IPS modules and the ability to use Identities in access-lists that could arguably called NGFW. More recently they had the CX module (now Approaching End of Sales). It had several NGFW features including AVC, Web Security Essentials (WSE) and IPS.
The current product lineup include the FirePOWER modules with technology acquired from Sourcefire being developed and integrated into the Cisco security portfolio, including ASAs. Those also have AVC (basically the ability to look deep into a flow and determine application-specific (or even "microapplication") information. You leverage that with the addition of IPS, Web filtering and/or Advanced Malware Protection (AMP) licenses on the FirePOWER modules.
The advantage is that you are able to protect your enterprise from modern-day threats. With the vast majority of malware being exploits from web pages (or at least carried over http/https), the traditional firewall with a rule allowing, say, only http from inside clients does nothing to protect against those threats. Client side anti-malware software can help, but it may be too late once the malware has been identified.
