qubit

The smallest unit of Quantum Information i.e. a qubit stores a linear combination of the bases as opposed to 0 or 1 for a classical bit
The act of measuring a qubit yields either $| 0 ⟩$ or $| 1 ⟩$ with some probability.
A qubit can contain infinite amounts of information since the binary expansion of $θ$ and $ϕ$ may not be terminating. However, a measurement of the qubit will make the state collapse to $| 0 ⟩$ or $| 1 ⟩$
Consider:$$\begin{align}|\psi\rangle &= \frac{1}{\sqrt2} |0\rangle + \frac{e^{i\phi}}{\sqrt2} |1\rangle \ &= \frac{1}{\sqrt2} [|+\rangle + | - \rangle] + \frac{e^{i \phi}}{\sqrt2}[|+ \rangle - |-\rangle] \ &= e^{i\phi/2} [\cos \frac{\theta}{2} |+ \rangle - i \sin \frac{\theta}{2} |-\rangle] \end{align}$$This now allows us to measure the relative phase

Classical: 00, 01, 10, 11
Quantum: $| 00 ⟩ = | 0 ⟩ \otimes | 0 ⟩, \dots$
Not all two qubit states can be written as a product of two single qubit states
For a $n$ qubit system, we have $2^{n}$ bases and therefore require $2^{n}$ complex coefficients. However, measurement only reveals $n$ bits of information.
If a state is expressed in computational basis i.e. ${| 0 ⟩, | 1 ⟩} := α | 0 ⟩ + β | 1 ⟩$ one can get information about $| α |^{2}$ and $| β |^{2}$ but not their relative phase. A measurement in the diagonal basis can yield information about relative phase as well.

Action of a unitary operator on a single qubit

After the action of a unitary operator each state on a bloch sphere goes to another point on the bloch sphere
operations are rigid body rotations and reflections
Any 2x2 unitary matrix can be written as: $$\begin{align} U &= e^{i\alpha} \exp(-i \theta \hat{n} . \vec{\sigma}/2 ) \ U &= aI + b \sigma_x + c \sigma_y + d \sigma_z \end{align}$$