quantum mechanics - Confusion about ket states and bra with position

I am very confused about the bra-ket notation of states and the fact that $$\psi(x) = ⟨x|\psi⟩$$ and $$⟨x|x'⟩ = \delta(x-x')$$ are true. What does this mean? What is the ket $|x⟩$, is it just some kind of identity vector? And what even is a state $|\Psi⟩$. What does it look like, is it an infinite vector where $\Psi_n$ is just that wavefunction with the principle quantum number being $n$?.

E.g. the solution to the infinte square well is given by the wave function $$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n \pi}{L} x\right),$$ if I am not mistaken. Then, can one say that $$ \psi_1(x) =\sqrt{\frac{2}{L}}\sin\left(\frac{1\pi}{L} x\right) $$ $$\psi_2(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{2 \pi}{L} x\right)$$ and etc.? And if so, how is that related to $⟨x|\psi⟩ = \psi(x)$?

Answer

Let me do a comparison with the finite dimensional case. A vector $v$ is an abstract entity belonging to a finite dimensional Hilbert space $\mathcal{H}$. Now in order to make actual computations with it, we usually handle with its components $v \to (v_1,v_2,\ldots)$. Once we choose an orthonormal basis the components are just the scalar products with the basis vectors $v_i = \langle v| e_i\rangle$. They are thus numbers because the scalar product provides a map $\mathcal{H}\times \mathcal{H} \to \mathbb{C}$.

In the infinite dimensional case the label $i$ of $e_i$ can assume infinitely many values. In this particular case it is actually continuous: $$ i \to x\,,\quad e_i \to |x\rangle\,. $$ And so the wave function $\psi(x)$ it simply the "$x$th" component of $|\psi\rangle$, $$ v_i \to \psi(x)\,,\quad\langle v | e_i\rangle \to \langle \psi | x\rangle\,. $$ As I said before this requires the basis $|x\rangle$ to be orthonormal and when $x$ is a continuous parameter the condition to be imposed is with the Dirac $\delta$. Moreover we must also impose that it is complete, namely $$ \int dx\,|x\rangle \langle x| = \mathbb{1}\,, $$ which is an operator equation. We can use this property to compute scalar products explicitly $$ \langle \psi|\chi\rangle = \langle \psi|\mathbb{1} |\chi\rangle = \int dx\, \langle\psi|x\rangle\langle x|\chi\rangle = \int dx \,\psi(x)\,\chi^*(x)\,. $$ In the same way as we would compute $\langle v | w\rangle = \sum_i v_i w^*_i$.