Well, as you may already know, "the Internet" is the term we use to talk about the worldwide, interconnected network of computers and also the data that travels among them. The first connection on what would become the Internet happened in 1969, when computer operators at Stanford and UCLA connected two computers over a network.

Computers of course existed well before that; there's some argument over what constitutes the first computing device, but broadly we can say that ENIAC, operational in 1946, was the first general-purpose electronic Turing-complete computer.

ARPANET, the Advanced Research Project Agency Network, was what the Stanford-UCLA connection helped launch; it was meant to solve the problem of computing resources being located away from researchers who needed computing time to solve problems. ARPANET existed until 1990, when it was shut down and control of the network was passed to the NSFNET, the National Science Foundation Network, which had existed as a portion of ARPANET starting in 1985. NSFNET, importantly, initially forbade commercial traffic on its network, which led to tensions in the internet community, and led eventually to it being decommissioned and replaced by the current commercial Internet. (I am oversimplifying greatly here, but you asked for broad strokes.)

The important thing to keep in mind is that traffic on the early internet was not web traffic; no dank memes and kitteh pics were circulating. Email was introduced in 1971, and Dr. Larry Brilliant (his real name) created primitive conferencing software in 1981, which led to the launch of The Well in 1985. There were also multi-user games on the internet as early as 1978, the first one being MUD; and Usenet launched in 1980.

But what most people think of when they think of "the Internet" is the visual internet we access through a browser, which is more properly termed the World Wide Web. The WWW is the brainchild of Tim Berners-Lee, who wrote the original hypertext markup specification and server software to deliver results over the Hypertext Transmission Protocol (HTTP) in 1989 to solve problems related to document retrieval at CERN; you can still see the very first web page, preserved here; it was put online in 1991. Berners-Lee's HTML specification has been built upon over the years, and has also led to the creation of Cascading Style Sheets (CSS) for document style and ECMAScript (JavaScript) for interaction on pages.

But the actual question you were asking is when did the Internet become common? Old timers on the Internet still talk about the "eternal September," when in 1993 AOL offered free Usenet access to its subscribers. Generally in the period before that, new students would get internet access at their universities, and the period of acculturation to online communities that started in September was fairly predictable to people who were established in communities; 1993 is seen as the year that new users were being added constantly, rather than in waves. 1993 is also when we get the Mosaic browser, the first graphical internet browser, which is often credited with increasing the popularity of the Web. So if I had to pick a yer, I would say 1993 is when it became "a thing," although other important dates in Internet history, such as the founding of Yahoo in 1985, the launch of Hotmail in 1996, and other things that sadly run us into the 20-year-rule are also important.