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I've heard and read about the passive sign convention, but don't really understand its meaning and importance in solving electric circuits. Is this because of conventional current which flows opposite electrons or is it something else?

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The passive sign convention is necessary to determine whether a circuit element is dissipating power, like a resistor, or providing power to the circuit. To calculate an element's power using P=VI you have to measure the voltage across the element. Suppose you use a voltmeter and you connect the red lead to one terminal of the element and the black lead to the other terminal...by doing this you have designated the terminal with the red lead as the posiitive "+" terminal. The passive sign convention says that you must measure the current entering the positive terminal and multiply by the voltage measurement to calculate the element's power. If the result is positive the element is dissipating power, if it is negative the element is supplying power to the circuit.

Of course, the choice of where to put the red lead is arbitrary. But if you reverse the connections and measure the current into the other terminal, then the signs of both the current and voltage will change and their product will still have the same sign. If current is actually flowing out of a terminal then the equivalent current flowing in to the terminal is a negative value.

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The "passive" in passive sign convention means that you are assigning positive voltage drops to passive components (i.e. loads). Negative voltage drops then occur across sources.

Why you ask? Why not positive voltage drops across sources and negative voltage drops across loads (i.e. active sign convention which you will probably never see because no one uses it, but it is equally valid).

I always assumed it was because it's more straightforward to work with positive numbers and there tends to be more load than sources in a circuit. So if you make voltage drops across loads more positive, you have more positive numbers in your equations.

I'd also like to point out that a negative voltage drop is actually a voltage rise which intuitively makes sense for a source.

I could have just as easily said that in passive sign convention, loads experience negative voltage rises and sources experience positive voltage rises. But we tend to talk in terms of voltage drops rather than voltage rises.

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