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 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 and FoldList:
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:
LinearRecurrence[{1, 1}, {1, 1}, 10]
{1, 1, 2, 3, 5, 8, 13, 21, 34, 55}
Hopefully these examples inspire you.