An AMI (Amazon Machine Image) is a template of an operating system like a CD of an operating system that you can install on any machine that you have similarly an AMI is an installation of an operating system that you can install on any servers which fall in the Amazon infrastructure.
Whereas Instances are nothing but the hardware machines on which you will install AMI. So as said AMIS are templates that can be installed on machines and these machines are called instances. Instances also have types based on the hardware capacity.