A qubit state is just a vector. You could write it as \(\begin{pmatrix}\alpha \\ \beta\end{pmatrix}\) every time, but that gets tedious. Dirac notation is shorthand that makes quantum math readable.
A ket is just a column vector with a name inside the \(|\ \rangle\) symbol.
$$|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \qquad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$
$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$$
That's it. \(|0\rangle\) is just a name for the column vector \(\begin{pmatrix}1\\0\end{pmatrix}\). Nothing deeper.
Take the ket, flip it to a row vector, and complex-conjugate each entry.
$$|\psi\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix} \quad\xrightarrow{\text{flip to row}}\quad (\alpha, \;\beta) \quad\xrightarrow{\text{conjugate}}\quad \langle\psi| = (\alpha^*, \;\beta^*)$$
Complex conjugate means: flip the sign of the imaginary part. So if \(\alpha = 3 + 2i\), then \(\alpha^* = 3 - 2i\). If \(\alpha\) is already real, nothing changes.
$$|0\rangle = \begin{pmatrix}1\\0\end{pmatrix} \quad\Rightarrow\quad \langle 0| = (1, \; 0)$$
$$|1\rangle = \begin{pmatrix}0\\1\end{pmatrix} \quad\Rightarrow\quad \langle 1| = (0, \; 1)$$
$$|\psi\rangle = \begin{pmatrix}\frac{1+i}{2}\\ \frac{1-i}{2}\end{pmatrix} \quad\Rightarrow\quad \langle\psi| = \left(\frac{1-i}{2}, \; \frac{1+i}{2}\right)$$
Put a bra and a ket together. It's just row × column matrix multiplication:
$$\langle x | y \rangle = \text{(row vector)} \times \text{(column vector)} = \text{one number}$$
$$\langle 0|1\rangle = (1, \; 0) \cdot \begin{pmatrix}0\\1\end{pmatrix} = 1 \cdot 0 + 0 \cdot 1 = 0$$
Zero → orthogonal → measuring \(|1\rangle\) never gives outcome \(|0\rangle\).
$$\langle 0|0\rangle = (1, \; 0) \cdot \begin{pmatrix}1\\0\end{pmatrix} = 1 \cdot 1 + 0 \cdot 0 = 1$$
One → identical states → measuring \(|0\rangle\) always gives outcome \(|0\rangle\).
If \(|\psi\rangle = \alpha|0\rangle + \beta|1\rangle\):
$$\langle 0|\psi\rangle = (1, \; 0) \cdot \begin{pmatrix}\alpha\\\beta\end{pmatrix} = \alpha$$
The inner product just picks out the amplitude for that basis state. Then:
$$|\langle 0|\psi\rangle|^2 = |\alpha|^2 = \text{probability of measuring } |0\rangle$$
Ket \(|\psi\rangle\) — column vector. The qubit state.
Bra \(\langle\psi|\) — row vector, conjugated. The "mirror" of the ket.
Braket \(\langle x|y\rangle\) — single number. Multiply row × column. Measures overlap.
For probability: \(|\langle x|y\rangle|^2\) = probability of getting outcome \(|x\rangle\) when measuring state \(|y\rangle\).
\(\langle 0|0\rangle = 1\) — same state, certain outcome
\(\langle 1|1\rangle = 1\) — same state, certain outcome
\(\langle 0|1\rangle = 0\) — orthogonal, impossible outcome
\(\langle 1|0\rangle = 0\) — orthogonal, impossible outcome
\(\langle 0|\psi\rangle = \alpha\) — picks out the \(|0\rangle\) amplitude
\(\langle 1|\psi\rangle = \beta\) — picks out the \(|1\rangle\) amplitude