AWS VPC - Internet Gateway vs NAT

Question

What is an Internet Gateway? What is a NAT Instance? What services do they offer?

Reading AWS VPC documentation, I gather they both map private IP addresses to internet route-able addresses for the outgoing requests and route the incoming responses from the internet to the requester on the subnet.

So what are the differences between them? What scenarios do I use a NAT Instance instead of (or beside) an Internet Gateway? Are they essentially EC2 instances running some network applications or are they special hardware like a router?

Instead of simply pointing to AWS documentation links, can you please explain these by adding some background on what is public and private subnets so any beginner with limited knowledge of networking can understand these easily? Also, when should I use a NAT Gateway instead of a NAT instance?