Voltage is a difference of potential energy for electric charges, and potentials are defined from forces, so that $F=-\nabla V$, where $V$ is the potential and $F$ is the force. When you have determined the potential $V$ you can now add any constant you want, or any function $f$ that doesn't depend on coordinates, because $-\nabla V=-\nabla (V+f)$, as $f$ doesn't depend on coordinates. This last thing is what's known as setting an origin of potential energy, something you sure have heard about. Well, in that function $f$ where you set your origin for potentials, you can have negative voltages between two points.

Another way to see the voltage is the work you must do per unit charge to move that charge from one point to another, even here, when we're dealing with differences, you can also have negative voltages.

If you're talking about curcuits, all of the above applies, you can set a potential difference $V$ by, let's say, a battery, then you can use some device so that the potential energy is even lower than the one set by the - sign of the battery, as you set the potential origin 0 for that battery, then that point will have a negative voltage.