Sometimes a Pi just won't boot. Or it'll boot, but it'll do weird things. Or you don't have an HDMI display, and you can't log into your Pi via SSH. Or maybe you're like me, and someone 'accidentally' cut your Raspberry Pi in half, and you want to see what it's doing since it won't boot anymore.
The Raspberry Pi can output information over a 'serial console', technically known as a UART (Universal Asynchronous Receiver/Transmitter). Many devices—including things like storage controller cards, which in a sense run their own internal operating system on an SoC—have a 'UART header', which is typically three or four pins that can connect over the RS-232 standard (though many do not operate at 12v like a traditional serial port! Use a USB-to-TTL adapter like the one I mention below).
Simply Embedded has a great overview of UART if you want to learn more.