What are waves? I am layman when it comes to physics, but I cannot understand what is often meant when waves are discussed.
I understand physical waves, e.g. water, as a type of behaviour of some fluid or medium. So in this picture there's a bunch of molecules moving in a certain fashion.
But I fail to understand the concept of waves when is talked about radio waves or even worse when I read about the double slit experiment and particle wave duality. What are these wave made off / what medium do they live in?
 A: Ever since the discovery of calculus mathematics, mathematical functions are used in order to model observations and special axioms are introduced  in order to connect the pure numbers of mathematics with observable numbers.
Waves in water, the first waves to be observed are defined by their height and frequency in time and motion in time and space, and can be fitted with the solutions of what is called a "wave equation"

The wave equation for a plane wave traveling in the x direction is $$\frac{\partial^2y}{\partial x^2}=\frac1{v^2}\frac{\partial^2y}{\partial t^2}$$ where $v$ is the phase velocity of the wave and $y$ represents the variable which is changing as the wave passes.

The solutions are in terms of sines and cosines which have the frequency of the repetition  inherent.
When Maxwell brilliantly united what were disparate laws and equation in the study of electricity and magnetism into one set of interdependent differential equations, to everybody's surprise, a wave equation resulted.  The solutions of this fitted light and the rest of electromagnetic phenomena to very great accuracy.
Here is a polarized electromagnetic wave showing that it depends on Electric and B magnetic fields changing in time.


Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right. The electric and magnetic fields in such a wave are in-phase with each other, reaching minima and maxima together.

From infrared radiation to visible light to radioto  gamma rays, this model is always validated .

even worse when I read about the double slit experiment and particle wave duality.

The waves in particle wave duality are probability waves as seen here. The wave function postulate is in the third page of the link. This means one must accumulate events in order to see the interference patterns predicted by the wave nature of the probability of interaction See this answer of mine on this..
A: You probably realize this is a very general question and a wall of text can be insufficient to answer it thoroughly. Still, one the most concise definitions of waves is that it's an oscillation which propagates in space.
When you ask "what are these waves made of", I suppose you wonder about a physical quantity which oscillates in each particular case. For example, in case of waves on the water surface such quantity would be the vertical displacement of the molecules of water. Generally the answer depends on the nature of waves. In electromagnetic waves, it's the electromagnetic field. In de Broglie's waves, it's the probability amplitude of finding a particle.
Regarding the medium where the waves can propagate, it again differs between different waves. For example, electromagnetic waves can propagate in vacuum while acoustic waves cannot.
A: A wave is simply a function of space and time $F(\vec r, t)$ such that the value of $F$ at any point $\vec r$ and time $t$ depends on its values at nearby points and times in a deterministic way that is described mathematical by an equation called the wave equation.
What does the function $F$ represent ? Well, that depends on the type of wave. For waves on the surface of a pond, $F$ represents the height of the water. For sound waves, $F$ represents the pressure at a particular point and time. For electromagnetic waves, $F$ represents the values of the electric and magnetic fields - in this case, $F$ is not just a single value at each point in space and time, but is actually a pair of vectors $\vec E(\vec r,t)$ and $\vec B(\vec r, t)$. And for a quantum wave function in position space $F$ is a complex number that is related to the probability of observing a particle at that point and time (there are other types of quantum wave functions where $\vec r$ is not a position in physical space, but is a position in a more abstract space such as momentum space).
Although these waves are models of very different physical phenomena, they are all based on the same underlying mathematical structure.
A: I assume that your question is about electromagnetic waves and light. This the same question that Huygens asked himself about light in the 17th century. He noticed that sound does not travel through vacuum whereas light does. Newton's solution we was that light consists of particles. This however does not explain the wave behavior of light. The present main stream physics answer is that light consists of particles, photons, the behavior of which is governed by electromagnetic waves. These waves are believed to have no medium.
A: Broadly, a wave is simply a quantity that changes in a periodic way over space and/or time.
In the case of a water wave, the changing quantity is the distance through which the surface is displaced- or, if you like, just the height of the surface.
With a wave on a string, the changing quantity is the displacement of the string from its resting position.
With a sound wave, the changing quantity is air pressure.
The frequency of a wave is the number of times per second that the changing quantity oscillates. The wavelength is the distance between successive occurrences of maximum or minimum values of the changing quantity.
With electromagnetic waves, the changing quantity is the strength of the electric and magnetic fields that permeate space. If you are happy with the idea of an electric field, for example, then you can think of the wave as being a periodic fluctuation in the strength of the field at every point.
All particles seem to have a wave-like aspect (the particle wave duality you mentioned). In the case of those waves, the changing quantity, very loosely speaking, is a measure of the probability that the particle will be found at a particular point.
Strictly speaking, the idea of a pure wave with a precise frequency and wavelength everywhere in space is a bit of a fiction. In reality waves tend to be less uniform than that, and their properties can become very difficult to model precisely.
A: For the layman: The defining characteristic of a wave is that it takes time for an influence to travel from place to place. E.g. Throw a stone in the water, and the influence (ripples) takes time to travel across the pond.
Physics deals with abstract entities where the medium is not always physical, from water waves (medium = water) to light and radio waves (medium = Electromagnetic field) to probability waves in Quantum Mechanics (where the medium may only exist in an observer's head)
Fun Fact: Newton's original formulation of Gravity had gravity affecting objects instantaneously (no time delay) so was not a wave theory
