I assume that you know postulates of quantum mechanics. In case you don't know or you are not sure, revise from my lecture notes on moodle.

The postulates have been given in an abstract setting. There is no clue as to how computations can be done. The canonical quantization rule, or any other equivalent rule, which gives commutation relations are very powerful and contains all the necessary information for computations of measurable quantities and making predictions which can compared with experiments. Click

I believe that you will agree that it is very convenient to work with familiar realizations, rather than abstract objects, and for which you already know the rules of manipulations. It is indeed possible to describe the  states in terms of functions, or column vectors consisting of a few components with complex entries. Similarly, the  dynamical variables, which are to be described by hermitian operators in some abstract space, will be replaced by differential and other familiar operators acting on functions. In case the state vectors are  $n$- component column vectors, the operators would be $n\times n$ matrices.


You may ask how do we make this transition from abstract to concrete language? What information and tools are needed for this switch  over?  Mathematically one needs to select a basis and work with a suitable representation. This will require a lengthy  preparation, which is very important for understanding the subject fully.

However, we shall bypass most of these mathematical details, and only mention those aspects which are indispensable  for our limited purpose of understanding representations as fast we can, in the process making minimum compromises.
Here a close contact with physical intuition will the guiding principle.

The price to be paid is that several details will be skipped and only results will be summarized. A few results will be  explained and used without a proof.

Goals 

The objective of this lecture is to explain the following.

1. All possible states of a particle (in one dimension) can be described by square integrable complex valued functions $\psi(x)$ of position. Square integrability means \n       $$ \int_{-\infty} ^\infty |\psi(x)| ^2 dx < \infty. $$
2. This is the so called  coordinate representation, also known as the Schrodinger representation.
3.  In the coordinate representation, action of the position and momentum operators is given by   $$  \hat{x} \psi(x) = x\psi(x), \qquad \hat{p} \psi(x) = -i \hbar \frac{d\psi}{dx}. $$
4. The operators for other dynamical variables, functions of $x$ and $p$, are obtained by replacing $x,p$ by the corresponding operators.
5. Very many different representations are possible.

Milestones

1. Analysis of position and momentum thought experiments leads to the  Heisenberg uncertainty principle. In general not all dynamical variables can be measured simultaneously with arbitrary accuracy.
2. Quantum theory is not deterministic, and only probabilities of outcome of an experiment can be predicted.
3. While probabilities are measurable quantities, the set of probabilities for all possible outcomes does not provide a complete description of  states of a quantum system. A complete description of state of a system is provided by the corresponding probability amplitudes.
4.  The  commutation relations determine properties of operators,  and hence the action of the operators in the vector space of all possible states can be determined.

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