This article describes subnets mostly based on IPv4, but is in principle very similarly applicable on IPv6. It is recommended to have a look at network addresses to understand this article.
The internet is a lot of computers or devices connected to a network. The whole of these networks is called the internet. Every network can consist of partitions of if self. Especially the internet could be viewed as such a vast network, consisting of many subnets. This makes managing this vast number of computers a lot more easy, as there can be many people, institutions and companies be responsible for a fraction of the whole and manage and administrate it on their own.
Hosts on subnets may communicate freely within their own subnet, but require a router to be able to communicate to another subnet. Without a router you cannot reach any other network. On the internet, those routers are mostly part of the infrastructure of ISPs and their carrier. They use so-called routing tables, that so to say know where a package is to be sent to, if it is not meant for the own network. At home, this infrastructure part is the router, that connects your home network with the rest of the world. So essentially it makes you a part of the internet, but you have your own little atoll in this ocean of atolls and computers. So the router is only used, when communication is supposed to go outside. It sends outgoing communication to into the network of your ISP. From there on, it's the job of others, to see that it reaches its destination.
In the IPv4 sense, the subnet is described by a part of the network address. The other part is the host. When you have a look at the IP address 192.168.0.1, you cannot really say, what part of it is part of the network number and what is part of the host number. This is where the subnet mask comes into play.