I'm really surprised if this question isn't a duplicate, but since I failed to find one that asked about the Fibonacci sequence rather than someone using it as an example, I'll answer.

The most natural approach, besides using the built-in [`Fibonacci`](http://reference.wolfram.com/mathematica/ref/Fibonacci.html) function, recursion:

    f[0] = 0; f[1] = 1;
    f[n_] := f[n] = f[n - 1] + f[n - 2]  (* note memoization *)

    Array[f, 10]

>     {1, 1, 2, 3, 5, 8, 13, 21, 34, 55}

Better performing may be [`Fold`](http://reference.wolfram.com/mathematica/ref/Fold.html) and [`FoldList`](http://reference.wolfram.com/mathematica/ref/FoldList.html):

    fibonacciList[n_] := Module[{x = 1}, Rest @ NestList[x + (x = #) &, 0, n]]

    fibonacciList[10]

>     {1, 1, 2, 3, 5, 8, 13, 21, 34, 55}

Another useful way uses [`LinearRecurrence`](http://reference.wolfram.com/mathematica/ref/LinearRecurrence.html):

    LinearRecurrence[{1, 1}, {1, 1}, 10]

>     {1, 1, 2, 3, 5, 8, 13, 21, 34, 55}

Hopefully these examples inspire you.