If you are using it as a noun, you use the article "the" and you capitalize "Internet."
"The Internet began in the early 1980s." "IPv6 is now being adopted throughout the Internet." "Italki is a website on the Internet."
In English, we often use nouns as adjectives. You can use the word "internet" as an adjective, meaning "present on the Internet" or "using the Internet" or "enabled by the Internet. In this case, the word "internet" does not need "an" or "the."
Warning! Usage is evolving! When used as an adjective, some people still capitalize it, "Internet bookseller," and some do not, "internet bookseller." In these examples, I will not capitalize it.
"You can often find higher interest rates at internet banks than you can at brick-and-mortar banks."
"On October 22nd, internet access was crippled in parts of the United States; it was caused by a DDOS attack from a botnet running, in part, on internet-enabled appliances like DVRs."
I'm sorry if this is confusing: we might need "an" or "the," not because of the word "internet," but because of the word that "internet" modifies. You can tell because we would still need the article if we left out the word "internet."
"Google, Bing, and Baidu are internet search services." No article needed.
"Bing is a search service." We use the article "a" because there are many search services.
"Bing is an internet search service." We change "a" to "an" because of the spelling-and-euphony rule. We use "an" because we are talking about "a search service," not because we are talking about the Internet.
"Baidu is the search service most commonly used in China." We need the word "the" because we are thinking about one specific search service.
"Baidu is the internet search service most commonly used in China." We need it here for the same reason.