For a computer or other device, being offline means it is not connected to a network or the Internet. While a device is offline, it cannot send or receive data over the Internet, and applications that require an Internet connection may not function properly. A computer may go offline due to a technical problem with itself or other network devices. A computer's user may also choose to go offline intentionally to prevent it from sending and receiving data.
Being offline may also more broadly refer to a device not being powered on, connected to other devices, and ready to be used. For example, when you try to print to a printer that is not plugged into power, you may get an error message saying that the printer is "offline." This does not mean that the printer needs an active Internet connection, but instead that it is simply not in a functional state.
Some applications expect a constant connection to the Internet to check for new content or sync data to the cloud in the background. However, if you know that you won't be online for a while, some apps give you the option to "work offline" in an offline mode. Offline modes disable any functionality requiring an Internet connection until you choose to go back online. For example, setting an email client to work offline will stop it from automatically checking for new messages in the inbox; you can still write a message and send it to the outbox, where it will wait until you are back online.
NOTE: Prior to the widespread adoption of always-on Internet connections (like DSL, cable, and mobile networks), you needed to dial-in to your ISP to actively connect to the Internet. This tied up your phone line, so you would end the call and go offline when you were finished using the Internet.