Wave

Since you have particularly asked to explain Doppler's effect "in simple language", I will avoid all the mathematical formulas and explain the basic physics in a few lines.

Imagine we are standing away from each other by some distance and we are holding a straight horizontal spring of infinite length.

If either of us moves towards the other, the part of the spring between us will contract. Its length will be shorter but the number of 'loops per unit length' of the spring will increase.

Now, replace the spring with some wave signal (mechanical, electromagnetic or gravitational). A similar thing will happen. The wave will appear to be compressed, its wavelength will appear shorter, but the number of cycles per unit time will increase, thus an apparent increase in frequency will be observed. The reason for using the term 'apparent' for the wave is that the wavelength and frequency of the wave observed by you will be different from what I emit.

If either of us moves away from each other, the opposite effect will happen. The length of the part of the spring between us will increase and the number of loops per unit length will decrease. Hence if we replace the spring with a wave signal, its wavelength will increase but the apparent frequency will decrease. 

Now for the effect of the medium; imagine you have a friend who is trying to pull the string from me towards you. Then, to you, the spring will appear to be compressed. If he tries to pull the spring away from you and towards me, the part of the spring will appear to be elongated to you. In this case, a similar wave analogy can be applied.

I hope this helps.