As shown in the Bloch Sphere there are 3 axis on which a qubit could be flipped mathematically.
Will rotate the phase 180° around the X-axis, between |+⟩ and |-⟩.
Given the mirror rotation it will flip the probabilities between |0⟩ and |1⟩.
X = | 0 1 |
| 1 0 |
Hence the gate will flip the qubit as shown
X|0⟩ = |1⟩
X|1⟩ = |0⟩
Will rotate the phase 180° around the Y-axis, between i|0⟩ and -i|1⟩.
Given the mirror rotation it will also flip the probabilities between |0⟩ and |1⟩, while also introducing a phase shift.
Y = | 0 -i |
| i 0 |
Hence the gate will flip the qubit as shown
Y|0⟩ = i|1⟩
Y|1⟩ = -i|0⟩
Will not rotate 180° around the Z-axis, between |0⟩ and |1⟩. This is a key difference between the X and Y gates. The qubit will only phase shift on the Z-axis, and only flipping the phase for the |1⟩ state.
This will keep the probabilities between |0⟩ and |1⟩ the same but produce a phase shift along the |+⟩ and |-⟩ (x-axis).
Z|0⟩ = |0⟩
Z|1⟩ = -|1⟩