"What are the differences [x]b , [x]b',[x]s ?"

In brackets we give coordinates with respect to a certain basis.

For example, if the basis is B = { (1,-1), (2,1) } then an example of a coordinate in this basis is:

[ (3,2) ]B >>> 3*(1,-1) + 2*(2,1)

It tells you how many of each basis vector to add.

B' is just some other basis. We could call it anything.

The standard basis in R^2 is S = { (1,0), (0,1) }

We can write coordinates in the standard basis as either [x]S or simply x.

Now let's say we have a different  basis, B' = { (2,-1), (1,2) }. How would we write the above vector in this basis?

We have:

[ (3,2) ]B           >>>             3*(1,-1) + 2*(2,1) = (7,-1)  = [x]S = x

So the coordinates in B have been written in the standard basis.

Now we can work out what the coordinates are in the basis, B' :

x = (7,-1) = 3*(2,-1) + 1*(1,2)

[x]B' = (3,1)