Basic NAT is not a real firewall.
Basic NAT devices are not real firewalls, but they are usually considered 'good enough' for most home networks. By not forwarding requests or probes that originate from the internet to your LAN, a NAT device blocks most mischief.

A simple NAT device can not keep hackers from running DOS (Denial Of Service) attacks on you, but individuals rarely get attacked like that. It will keep out people looking for file shares, rogue mail servers and web servers, and most port based exploits. Most also protect against SMURF and WinNuke attacks.

With a NAT device and a good anti-virus program, you should be safe from the most common kinds of internet attacks.

Some NAT routers have an advanced form of firewall built in that does 'stateful packet inspection'. This allows the NAT devices to filter out specific kinds of data on your router like SYN flood attacks, IP Spoofing, Teardrop attacks and others.

SPI is a general term that can describe a router that filters more kinds of attacks than basic NAT by closely examining packet data structures.
Of course, each manufacturer will implement different kinds of SPI so not all SPI routers are equal. Routers with SPI can often log attacks.