It depends on the coordinate system you're working in.
In a right-handed coordinate system (eg. x right, y up, z points toward the viewer), the right-hand rule applies, as mklingen describes in the existing answer.
In a left-handed coordinate system (eg. x right, y up, z points away from the viewer), the left-hand rule applies - you point your left thumb along the rotation axis, and a positive rotation turns in the directions your fingers curl.
So it's important to know what coordinate system you're working with. Unity for instance uses a left-handed coordinate system. 3DS Max uses a right-handed coordinate system. Changing handedness flips the sign of an odd number of axes, and all rotation angles.
"Oh, so Quaternions are basically... 4D vectors, then? That we just give another name?"
Not quite. Like 4-vectors, they are 4-tuples of numbers, but interpreted such that three of the four components are measured in imaginary units. That means that two quaternions multiply together differently than they would if composed of real numbers (because 1i * 1i = -1). This weird multiplication means that they combine like 3D rotations do (something you don't get out-of-the-box with real numbers), which is why we often use them to describe rotations and orientations.