Understanding The Bloch Sphere
In other words, the state space of a single qubit is the set of unit vectors in the two dimensional complex vector space $\mathbb{C}^2$ with basis vectors $|0\rangle$ and $|1\rangle$.
Operations on a single qubit are linear transformations which preserve the norm and therefore correspond to $2\times 2$ unitary matrices $U \in U(2)$.
The vector space $\mathbb{C}^2$ is a four dimensional real vector space, and the constraint $|a|^2 + |b|^2 = 1$ means that the state space of a qubit can be identified with the three dimensional sphere in four dimensional space $S^3 \subset \mathbb{R}^4$. Concretely, if $a = x + yi$ and $b=z + wi$ then $|a|^2=x^2+y^2$ and $|b|^2 = z^2 + w^2$ and so the state space of a qubit consists of the points $(x,y,z,w)\in\mathbb{R}^4$ satisfying:
The Bloch Sphere is a projection of the state space onto the more familiar two dimensional sphere in three dimensional space $S^2 \subset \mathbb{R}^3$. Importantly, under this projection unitary transformations of the state space correspond to ordinary rotations of the sphere. Furthermore, there is an explicit formula that determines precisely which rotation corresponds to a given unitary matrix. This makes the Bloch Sphere an indispensible tool for analyzing single qubit operations.
