Hi,
From your description it sounds like you are measuring one end to the ground.
Consider this a resistor:
A ---====---- B
and Gnd is (-) of the circuit
it sounds like you are measuring A-Gnd and B-Gnd. The true voltage drop across the resistor is the voltage difference between A and B.
A resistors is a basic component that can be junctioned with another component, say another resistor, in either parallel (where both leads of the resistors are connected) or in series (only one lead is connected to the other's with no connection from another component inbetween).
If we assume you make only a single line connecting the resistors in series, you will get voltage drop, current stays the same and follows Ohm's law and the resistors' values are constant. while in parallel, you will get 2 resistors making up 2 branches, in this case voltage stays the same on both but the current will divide.
The division of voltage follows "Voltage divider rule" and current follows "Current divider rule" (check wikipedia)
Now when the current passes through the resistor while it has voltage drop on it, there appears the term "power". Power is the amount of work/energy dissipated by the device (resistor) and it simply equals [Voltage drop on the resistor X That current passing through resistor] and it appears as heat in our physical world.
Here comes our decision, what is the most proper power rating to choose for that particular resistor to work well without getting burned and overloaded so fast ? commercially, we have 0.25, 0.5, 1, 2, 5...etc watts...pick one.
In real world, many things behave like a resistor, so we use modeling to represent them.