What is the difference between a scalar and a vector field? Could someone please indicate precisely the difference between a scalar and a vector field? I find no matter how many times I try to understand, but I always am confused in the end. So what exactly makes them different?
 A: A scalar field is simply a single function of, say  $n$ variables.  
Temperature is an example of a scalar field.  Temperature is a function of three variables that define position in a spatial coordinate system.  We can measure the temperature $T$ at each point $(x,y,z)$ and thus form a function $T(x,y,z)$.
A vector is a set of functions of $n$ variables.  
The electric field is an example of a vector field.  The electric field is actually comprised of three functions of three variables.  We measure the three components of electric field $\vec E$ along the coordinate axes at each point $(x,y,z)$ and thus "create" three functions $\vec E(x,y,z)=\hat x E_x(x,y,z)+\hat y E_y(x,y,z)+\hat z E_z(x,y,z)$.  Thus, the electric field has both a magnitude and a direction at each point in space.
A: A scalar is a number, like 3 or 0.227.  It has a bigness (3 is bigger than 0.227) but not a direction.  Or not much of one; negative numbers go in the opposite direction from positive numbers,  but that's all.  Numbers don't go north or east or northeast.  There is no such thing as a north 3 or an east 3.
A vector is a special kind of complicated number that has a bigness and a direction.  A vector like $(1,0)$ has bigness 1 and points east.  The vector $(0,1)$ has the same bigness but points north.  The vector $(0,2)$ also points north, but is twice as big as $(0,2)$.  The vector $(1,1)$ points northeast, and has a bigness of $\sqrt2$, so it's bigger than $(0,1)$ but smaller than $(0,2)$.
For directions in three dimensions, we have vectors with three components.  $(1,0,0)$ points east. $(0,1,0)$ points north.  $(0,0,1)$ points straight up.  
A scalar field means we take some space, say a plane, and measure some scalar value at each point.  Say we have a big flat pan of shallow water sitting on the stove.  If the water is shallow enough we can pretend that it is two-dimensional. Each point in the water has a temperature; the water over the stove flame is hotter than the water at the edges.  But temperatures have no direction.  There's no such thing as a north or an east temperature. The temperature is a scalar field: for each point in the water there is a temperature, which is a scalar, which says how hot the water is at that point.  
A vector field means we take some space, say a plane, and measure some vector value at each point.  Take the pan of water off the stove and give it a stir.  Some of the water is moving fast, some slow, but this does not tell the whole story, because some of the water is moving north, some is moving east, some is moving northeast or other directions. Movement north and movement west could have the same speed, but the movement is not the same, because it is in different directions.  To understand the water flow you need to know the speed at each point, but also the direction that the water at that point is moving. Speed in a direction is called a "velocity", and the velocity of the swirling water at each point is an example of a vector field.
I think the only other thing to know is that in one dimension, say if you had water on a long narrow pipe instead of a flat dish, vectors and scalars are the same thing, because in one dimension there is only one way to go, forwards.  Or you can go backwards, which is just like going forwards a negative amount.  But there is no north or east or northeast.  So one-dimensional vectors are interchangeable with scalars: all the vector stuff works for scalars, if you pretend that the scalars are one-dimensional vectors.
If this isn't clear please leave a comment.
A: Simple example of scalar field is when you imagine your room temperature at any point in space. there are certain difference between temperature near the place you are(you are heating air), and others. So you can say that the temperature field is bigger/smaller near you/far away from you. 
In other words - you can make a mark in $(x,y,z)$ coordinates that temperature near you is 25 degrees, temperature a meter from you $(x+1,y,z)$ equals 24 degrees.
Simple example of vector field is when you do a instant picture of a room and see the tiniest particles going their way, so the velocity of them  - the vector of each particle will give you a thing you need - a vector field.In other words - if you are walking, you make the particles collide with you, therefore their speed and direction can change you can make a mark in $(x,y,z)$ coordinates that velocity of particle near you is 10m/s and the direction is $(x_1,y_1,z_1)$ , and meter from you particle has such direction $(x_2,y_2,z_2)$ and speed that equals 8 m/s.
