Step 1: Know the four quantum numbers.
Every electron in an atom is described by four labels. They are the principal number $n$, the azimuthal number $l$, the magnetic number $m_l$, and the spin number $m_s$. Each label tells us one fact about the electron.
Step 2: What does $n$ tell us.
The principal number $n$ tells us the main shell and the size of the orbital. A bigger $n$ means the electron is farther from the nucleus. It does not tell us the shape.
Step 3: What does $l$ tell us.
The azimuthal number $l$ is also called the shape number. It directly fixes the shape of the orbital. When $l=0$ we get an s orbital, when $l=1$ we get a p orbital, and so on.
Step 4: What do $m_l$ and $m_s$ tell us.
The magnetic number $m_l$ tells us how the orbital is turned in space, that is its direction. The spin number $m_s$ tells us the spin of the electron. Neither of these decides the shape.
Step 5: Match shapes to $l$ values.
For $l=0$ the shape is a sphere. For $l=1$ the shape is like a dumbbell. For $l=2$ the shape is more like a double dumbbell or clover. So the shape changes only when $l$ changes.
Step 6: Pick the right answer.
Since only $l$ controls shape, the answer is the azimuthal quantum number.\[ \boxed{\text{Azimuthal quantum number } (l)} \]