What does the taylor series give us ? To be more clear if we use the taylor series for x=2 it will give us an approximation of f(2) ? And why do we stop "adding" the f'''(a)/3! ...? Is there a rule that tells us when to stop ?
 A: It gives you an approximation of the function "close to" $x=2$.  The constant term is $f(2)$, the next term gives the local linear approximation, and later terms make an even better fitting polynomial.  You stop when the expression is accurate enough for your purposes or when you get tired, whichever comes first.  The error term is the way to determine the first.  The second is left as an exercise.
If you look at  this Alpha plotit shows $y=\sin x$ and the first two Taylor series:  $y=\frac 12\sqrt 2$ (the horizontal line) and $y=\frac 12\sqrt 2 + \frac 12\sqrt 2(x-\frac \pi 4)$ (the straight line).  The straight line is clearly a better approximation.  If I added in the next term, you wouldn't be able to see the difference from the actual function.  If I expand the range to $[0,1]$ like this you can see what good the second order term does.
A: technically you never stop 
depends on what level of accuracy you want 
so if you are stopping at $f''(a)/2!$ that is just an approximation 
hope that makes sense
