  Quantum Theory is one of the pillars of modern science developed over the
last hundred years. In this review paper we introduce, step by step, the
quantum theory understood as a mathematical model describing quantum
experiments. We start with splitting the experiment into two parts: a
preparation process and a measurement process leading to a registration of a
particular outcome. These two ingredients of the experiment are represented by
states and effects, respectively. Further, the whole picture of quantum
measurement will be developed and concepts of observables, instruments and
measurement models representing the three different descriptions on experiments
will be introduced. In the second stage, we enrich the model of the experiment
by introducing the concept of quantum channel describing the system changes
between preparations and measurements. At the very end we review the elementary
properties of quantum entanglement. The text contains many examples and
exercise covering also many topics from quantum information theory and quantum
measurement theory. The goal is to give a mathematically clear and
self-containing explanation of the main concepts of the modern language of
quantum theory.
