One of the few things that audio engineers can agree on is that there is a huge misunderstanding of the terms "phase" and "polarity". They are often used interchangeably, which is fine among old hands who are using terms like "flip phase" because it's fewer syllables than "invert the polarity" and everyone in the room knows what they mean. But it's a good idea to not ever say phase when you mean polarity just to avoid confusion.
Most of this confusion arises because many mixing consoles have a button near the trim pot that will allow you to invert the polarity of the signal coming in. That means swapping the connection from pins two and three of an XLR or the tip and the ring of a 1/4" TRS connector. The problem is that it's usually a tiny button and it's often labeled "∅" which is the symbol for phase to save space.
Polarity is an electrical condition, DC actually. Think of reversing a nine volt battery to put the + on the - terminal. Phase is a little more complex and can be caused by a few different things. Delay is the first, two mics on the same source can pick up the sound at slightly different times. Then you can get a peak on one input and then the same peak on the other input slightly later. For transients like a snare hit that might not matter much, you'll just hear a little delay. For longer tones like guitar or vocal you'll get those sounds adding together at some frequencies and subtracting from each other at other frequencies. This is called comb filtering because it happens at regular intervals. Halfway between each peak is a null.
Complicating this matter is the fact that you can hit the polarity button on your desk and make things shift around because of their phase issues. Lots of engineers will play around with a pair of overhead drum mics this way. The results can be subtle or not so subtle but it's more of an artistic thing than actually trying to accurately correct phase issues.
Another way that trick is used is to help out a vocalist on a noisy stage. If you've ever seen live footage of a band where some one is singing into a pair of mics taped together this is what's going on. The vocalist sings into one mic and the second mic is set up exactly the same but with the polarity inverted. The first mic pics up vocal plus stage noise, the second mic picks up mostly stage noise and with carefully set gain structure the engineer can make the stage noise all but disappear.
Double micing guitar cabs takes advantage of phase differences as well. The subtle differences between two different models of mic in two different positions lets the engineer push the two faders back and forth to get constantly changing comb filtering from the phase differences. It's a far cry from some other phase issues where there's a hollow sound resulting. I've gotten to really like this method because it puts a wide palette of tones at my fingertips and the EQs are usually untouched. It's always the guitar players' actual sound but I can make it punchier or mellower as needed. I can even make a solo cut through the mix without actually making it louder by changing what frequencies cancel and add. It's not science, it's just feel at that point.
Delving a little deeper, crossover filters produce phase issues. Well, some of them do. Fourth and eighth order crossovers shift the phase so far that it's back to where it started, that's why you see them used so often, because no further correction is needed (those would be the ones with 24dB or 48dB per octave roll off when you look at the settings). Some crossover filters like a second order Butterworth will shift the highs by 180°. In small bookshelf speakers this was often corrected by simply reversing the leads to the tweeter. Other types shift the highs by other amounts and this can be corrected by applying another filter that corrects the phase shift without altering the frequency content.
Lastly, there is a huge misconception that EQ circuits introduce phase issues into your signal. While this is true, it's not really a bad thing. It's just the way that they work. Each slider or knob on an EQ creates a little (or a lot) of phase difference at that particular frequency and then when the signals from all the filters are recombined you get signal that has been altered tonally but not with respect to phase. There might be EQs out there that produce poor results due to cheap components or bad implementation, but it's not the basic theory that's at fault.
There have been hundreds if not thousands of other articles written on this subject my fellow Brethren of the Knob and Fader. So if this has piqued you interest, get out there and find them. Prosoundweb.com has collected a lot of them so that's a good place to start.