For example, spinner. Its centre point between our fingers is a fixed point, it doesn't move.

Dear Rqeeb, another example using a paper sheet: folding it many times from its initial rectangular plane shape, and putting the folded version over the original rectangular location, there is a "folded" point which projection fall down on its original plane position.  Gianluca

Suppose you have a tree and tie to it a certain length of a rope and the other end of the rope is tied to the neck of a goat, no matter the direction of the movement of that goat the position of the tree remain fixed

One of the most practical areas of Mathematics is the theory of differential equations.  it plays a vital role in Physics and Engineering.  Fixed point theory guarantees that these equations have solutions.  This is one very practical result that should be noted.

A curious meteorological application of the hairy ball theorem (which is related to the fixed point theorem)involves considering the wind as a vector defined at every point continuously over the surface of a planet with an atmosphere. As an idealization, take wind to be a two-dimensional vector: suppose that relative to the planetary diameter of the Earth, its vertical (i.e., non-tangential) motion is negligible.

One scenario, in which there is absolutely no wind (air movement), corresponds to a field of zero-vectors. This scenario is uninteresting from the point of view of this theorem, and physically unrealistic (there will always be wind). In the case where there is at least some wind, the Hairy Ball Theorem dictates that at all times there must be at least one point on a planet with no wind at all and therefore a tuft. This corresponds to the above statement that there will always be p such that f(p) = 0.

The hairy ball theorem of algebraic topology: For the ordinary sphere, or 2-sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one p such that f(p) = 0.

For example the logistic growth?

R.M. May Simple mathematical models with very complicated dynamics. Nature 261:459-467

British refer to Brouwer's theorem as the "cup of tea theorem"... that is quite explanatory :)

You can begin with a rotation of a circle, that is a transformation with a fixed point, and with a counterexample, the translation, without fixed point. Then you can use Brouwer's fixed point as a generalization of the case of the rotation, with the paper sheet example. Now you can pass to the contraction Theorem,  that is totally different, similar to a series of approximations. You can think the fixed point theorem as a way to hit a hidden target: in the case of the contraction, with the possibility to move closer; in the case of the theorem of Brouwer you need to know that the target (or the targets) are in a bounded, closed (and convex) area. Years ago I wrote a short tale with a sort of James Bond needing a briefing on some mathematical topics for a mission of infiltration, and the local Q  trying to prep him with the target examples... (Banach - closer and closer; Brouwer, in bounded etc. area).   ^_^; A bit cinematographic one. More pacific examples are the baker transformation, or in meteorology...

Two major areas of application of fixed point theorems are:

1. showing existence of solutions to differential equations and at times with provision of iterative algorithms to find/approximate the solutions

eg. Schauder's fixed point theorem

2. Showing existence of economic equilibria

eg. Kakutan's fixed point theorem and the Nash equilibrium

Just use a simple pin you find at every household and two (nearly) identically shaped pieces of cardboard, say cut them from a shoe box or a parcel. Put two pieces together so that the shapes match as perfectly as you manage and make a hole in both. Keep one piece fixed and start rotating the other one. All the points will changed their location except for the one you fixed by puncturing  two pieces of cardboard. You map (rotation) between two sets of points on two pieces of cardboard has the only fixed point. 

To explain the fixed point theorem, take a small balloon and puncture it with that same pin - in a while you nearly perfect sphere will shrink to something you can idealize as a "point" (even though the remains of punctured balloon will not in reality look as a "point"). :-)

  Consider the following situation (inspired by Abiodun's answer):One gets up from his/her bed in the morning and returns to the same bed in the evening (after daily activities, some possibly described by differential equations) ? Could the bed, in this scenario with some folklore,  be considered a "fixed point" ?

In some countries there is a habit of gathering with family and friends and exchange gifts: every one brings a present, presents are collected, and then each participant may draw one blindly. It may happen that you draw your own present. Such a fixed point is usually considered an undesired event....

Here is an easy situation where a drawing might make the concept clear.

Imagine a square of side L where you draw a continuous line from the left side to the right side of the square. (This is what one calls a "continuous function f" from the "interval" [0,L] into itself). Next, draw a straight line from the 'bottom left' corner to the 'upper right' corner of the square. Then the straight line will 'cut/intersect' the original continuous line at least once (or at more places). 

Such points when looked at from the bottom side of the square (call them "x") are said to be "fixed points" of the function "f", i.e., they satisfy "f(x)=x" (in other words, the places of intersection mentioned above are at equal distances to the "bottom of the square" and to the "left side of the square"). 

An easy drawing of this idealized situation should make clear the idea of "fixed points" of a function "f". Indeed, as some other colleagues may have mentioned, there are many not so idealized but practical situations where mathematics guarantees the existence of fixed points.  

A real life example is that, if we observe the rotation of a wheel its every point move except its center, so its center is a fixed point. 

Fixed Point: A point that remains invariant point under a given transformation. In real life problems a fixed point indicates the situation where a steady state condition or equilibrium is reached.

Invariant temperature-

·      The boiling point of Water

.      The freezing point of Water.

·      The triple point of a substance-the temperature at which substance exists in its solid, liquid and gaseous state all at the same time.

Another example, from an old mathematical conundrum: 

A man leaves his town at 8 am and arrives a man leaves at 8am and arrives at the village at the top of the mountain at 2 pm. He remains until next day, when he leaves at 8 am to go back the same way to his village and arriving  again at 2 am . Now there is a point on the road where he  passes at the same hour of the previous day. How to prove it?

I think I read it on a book by Martin Gardner.

Fixed point is a point which can not be moved. See attached  picture.

The example presented by Stefano is a nice one, but it actually involves what is called a coincidence point of two functions f and g: a point x such that f(x) = g(x). For instance if two continuous functions f, g on the unit interval [0,1] are such that f(0) = 0, f(1) = 1 and g(0) = 1, g(1) = 0 then the graphs  look like wiggly diagonals of the unit square. They must agree somewhere between 0 and 1.

A simple proof is this: g - f is another continuous function which has a positive value  (+1) at 0 and negative value (-1) at 1. Somewhere between, it must take the value 0. This is the "intermediate value theorem" of calculus.

With the proper interpretations, Stefano's claim is proved by this approach.

These are all great answers. for non-mathematicians. However, there may be more in some of them than expected ! Indeed, it was recently shown in the journal CRORR (also in the current project on fixed points on  RG), that at least for functions of the single variable, there exist in fact two characterizations (necessary and sufficient conditions) of the fixed point :  "primal" and "dual". What do (if anything)  the above answers yield when the dual formulation is applied ?

Many statements above are correct. I can tell that fixed point theorem is used not only in physics  but also in economics (see Arrow - Debreu theory of general equilibrium).

But remember that explanations should be to non-mathematicians! Functional spaces are too complex to explain in few minutes. I am thinking about simple discrete example. Consider a game of telling words (sometimes city or geographical names) that start on the letter that is the last in the previous. Words cannot be repeated. For example, a sequence  Antwerpen -> Nizza -> ... .-> Ankara... corresponds to mapping A-> N, N-> A,... A->A. It may happen that many words in some language start and end on the same letter, so that when we have used all other options, the only way to continue is using words like AnkarA, Addis-AbebA, AfricA, AmericA, etc. And this mapping A ->A will continue afterwords and A will be fixed point of this sequence (if we ignore that new words will end).

Mechanical continuous example is rotation of a wheel in the system moving with the object. Spinner (recently on market, was added), but think about central point (rotation axis) of wind mill (also to produce electricity).

It is sad, when they say that the 'The discrete version of the logistic equation is known as the logistic map'. Hopefully it will soon be revised.

http://mathworld.wolfram.com/LogisticEquation.html

http://mathworld.wolfram.com/LogisticMap.html

For non-mathematicians geometrically: Let's press a sheet on the table with a pencil or a pergole at one point. Let's then turn the sheet on the table. This is the rotation of the plane or a mathematically speaking a mapping of the plane into  itself. The point at which the pencil is pressed is a fixed point of  the rotation mapping . Abstract application are mentioned ( for instance in differential equations) in the above answers.

Several natural examples of fixed points have been given in this thread, but the reservoir is limited. I already mentioned coincidence points of two functions as a more general type of phenomenon. There are nice examples of this phenomenon, too

Consider a balloon, filled with air. Its highly stretchable material surface is comparable with a sphere. If you deflate the balloon, with absolutely all air removed, you have a rimpled flat object with two layers pressed together. There will always be two antipodal points of the balloon which are pressed together after deflating.

This is Borsuk-Ulam's Antipode Theorem valid for n-dimensional "spheres", n = 1,2,3,....

https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem

Fixed points of single-variable continuous Lipschitz functions have two equivalent (geometrically different) characterizations: "primal" and :dual" . (See the file attached.) The examples given in the thread seem to refer to the primal one. Could anybody describe a  real-life example of a  fixed point using the dual characterization of fixed points ?

Loosely speaking, the eye of a hurricane can be viewed as a "fixed point" by a non-mathematician. This is a relatively small area ("point") of calmness around which the storm rotates (some "function") causing devastation !For graphics see, e.g., many satellite images in Wikipedia.

The example I was given many years ago is indeed very simple and suggestive: take a map of your country and drop it on the floor. That's it!

By the Fixed-Point Theorem we can guarantee that, at any time, there exist two diametrically opposed places on our planet having exactly the same temperature. Actually, we can find two diametrically opposed places in our planet having the same temperature and the same pressure simultaneously (see Borsuk–Ulam theorem). The Brouwer Fixed-Point Theorem is also used to prove the existence of mixed Nash equilibriums in multiplayer games, although I am not sure that this is something you want to explain to a non-mathematician...

Fixed point of a mapping is an element if we apply the mapping on this point and we obtain the same point. In real world problem, small stone are the fixed points for the mapping=wheat thresher.