Simply the route a particular signal takes through a chain of equipment and/or electronic components on the way to its destination. When we think of signal paths in audio we are usually thinking about connecting different pieces of equipment together and routing some signal(s) through them. An example of this would be something like a microphone to mixer to speaker or recorder setup. The signal path has the signal from the microphone pass from the microphone through those (and potentially other) devices on the way to being recorded or amplified (or both). But there is also a signal path inside each piece of equipment. A mixer may be configured to route signals in different ways internally bypassing or utilizing different gain stages along the way to achieve different results. Effects processors often have highly configurable internal signal paths depending upon what they are doing.
Basic Signal Path Diagram