Let f:[a,b] -> R a continuous function, [a,b] an interval in R and Pn the set of polynomials of degree lesser or equal n.
Let {xi| i in {1,...,m} } a set of points in [a,b].
A polynomial p of Pn is said to interpolate f iff f(xi)=p(xi) for all i of {1,...,m}. This is the most common approximation by polynomials. Other approximations are rarely used.

If m <= n+1 then a p, which interpolates f, exists and if m >= n+1 then p is unique (but doesn't have to exist !), so you would always choose m = n+1.
Note that for increasing n and m=n+1 the solutions don't have to converge to f !
They converge only for special xi and surprisingly the point mustn't have all the same distance. They must be clustered around the ends of the interval, xi must be equal to a + (b-a) arccos(i/m)/pi.

A simple way to calculate the interpolating p is (for m=n+1) using the Lagrange's formula:

       __m  (x-x1) ... (x-x<i-1>)(x-x<i+1>) ... (x-xm)                  
p(x) = ||   ------------------------------------------ f(xi)
       i=1  (xi-x0)...(xi-x<i-1>)(xi-x<i+1>)...(xi-xm)
But there is a number of more clever formulas to calculate p.