When it comes to internet connectivity, an absence of the same leads to people saying that the internet connectivity is down. This condition is mostly temporary, unless an unforeseen incident causes it to go down permanently.
In the context of software, when the application or certain features within the software stop responding, then the software, or that feature is said to be down.
Example of usage – “He was trying to connect to the internet from his home office, but there was no success. When he called up the internet service provider, he was informed by the operator, that the internet services were down, and that they were working on a fix.”