Meanings
Early use of the phrase (circa the mid-1990s) refers to literally breaking the Internet connection of a single device. It also refers to breaking the World Wide Web or the Internet itself, with the implication that it would be impossible or implausible to do so in most ordinary contexts.
How to pronounce "break the Internet":
To cause a connection to the Internet, or the Internet itself, to malfunction.
To overwhelm a web server through organic user-driven demand, such that the server goes offline or is in danger of doing so.
To deliberately overwhelm a web server through a cyberattack such as a DDOS, such that the server goes offline or is in danger of doing so.
To do or say something ostentatious or controversial, such that the behavior causes a dramatic increase in traffic to a website or commentary on social media; to go viral.