I won't get into too many details, but I'll have a go at the two things you've mentioned.
1. The capacitor. When a capacitor is charged up and is not in any circuit, there will be a DC voltage across the two terminals, which will tend to stay there (except for leakage - but let's ignore that for the moment and assume we are talking about a high quality one).
If you want to measure the voltage, then you have to choose a reference point and measure the voltage with respect to that reference point. For a solitary capacitor, that means picking one of the contacts as the reference point and connecting your meter to that (usually the 'common' or black coloured lead). It is generally assigned a voltage value of 0v because if you put the other lead of your meter on the same point, it's obviously not going to measure anything! Now, having connected the 'common' lead of your meter to this reference point, you then take the other meter lead and touch the other contact of the capacitor - which will give you a reading on your meter of however many volts the capacitor is charged to.
BUT is the measured voltage positive or negative? The answer is - it depends on which contact you used as your reference point. Say we have 10v on the capacitor. If you connect the meter common to the negative terminal (using that as the reference point), you will measure the other (positive) contact as being +10v - but if you use the positive terminal as your reference point, then you will measure the other (negative) contact as being -10v.
All voltage measurements (not just for capacitors) are measured with respect to a particular reference point. (Often this is 'ground' or 'chassis' - but not always.)
2. Creating a voltage drop. In one sense - this is stating the process a bit 'back to front'. First you have to remember, there has to be a potential difference (voltage) from a supply (of whatever sort) for any current to flow. Once you have a circuit that connects across this potential difference, the voltage around the circuit has to go from the highest voltage of the supply to the lowest voltage of the supply - so, in fact, those 'voltage drops' are already going to happen - just exactly where and how much they will be across each component in the circuit is the fun bit!
Lets take a simple case: a resistor. A voltage drop 'appears' across it when a current passes through it. (How much is defined by Ohm's law: V=IR.) I think 'appears' is a better word than 'create' in this context for understanding - but electrical designers have a pretty firm grip on this stuff and might use the word 'create' - because that's what they meant to happen.