So far I learned that a qubit can be written as $| \psi \rangle = \alpha | 0 \rangle + \beta | 1\rangle$ with $|\alpha|^2 + |\beta|^2 = 1$ and reparametrized as $| \psi \rangle = cos( \theta / 2) + e^{i \phi} sin (\theta / 2)$ to visualize it on the bloch sphere: https://en.wikipedia.org/wiki/Bloch_sphere. Moreover, the source I am reading says that the probability to measure $0$ is $|\alpha|^2$ and the probability to measure $1$ is $|\beta|^2$.
Which role does the angle $\phi$ play since changing it does not influence the probabilities $|\alpha|^2$ and $|\beta|^2$ (due to $|e^{i \phi}| = 1$).
While thinking about this question it came to my mind that one could measure the qubit with different basis as well. Is this assumption true? If yes, are there some examples of application?