It's like a scope, but the vertical axis is either 1 or 0, not the actual voltage (like a digital signal), but you have more channels (at least 8, even in an entry level LA). This is really useful because even many serial protocols need more than two channels (SPI needs four), and you may also want to monitor something else (eg. input and output). Some (the streaming USB kind like the Saleae Logic and Usbee) have a larger memory depth (you can analyze longer streams of data, like a complete transmission). They also usually have more protocol decoding options (although this is also available in mid-end scopes for $$$). Even a basic LA will usually do SPI, I2C, Async serial (like RS-232) and 1-wire (used by the DS1820 temperature sensor). Most have more. Decoding means that the protocol is shown in human-readable form (eg. Read 0x42 or N-Ack, or the byte value for async), as opposed to just pulses.
You would typically use an LA if you have issues with the contents of your digital signal (eg. timing), or if a certain I2C slave sends an acknowledgement. You would use a scope for signal integrity issues like voltage levels, noise or ringing.