People often associate fractals with self-similarity and it is too often assumed that a fractal is just a figure that repeats itself to as one zooms in on it.

This definition of fractals, however, is inaccurate because not every figure which is self-similar is a fractal. The technical definition of a fractal is "a figure with a fractional dimension". In short, a fractal is a geometric figure which has a non-whole dimension. I wrote an article on fractional dimension which goes into this more.

What about a line? A line is self-similar - no matter how much you zoom into a line it looks like a line, but a line is not a fractal... it has a dimension of exactly 1.

Besides, how do you define "similar"? The Mandelbrot Set is a fractal and sorta looks like itself as you zoom in, but it doesn't repeat itself exactly. In fact, I've heard the Mandelbrot Set to be described as anti-self-similar (self-dissimilar?) because the figure has new designs the more you zoom in. The term "self-similar" really doesn't cut it for us mathematicians.

The Peaceful Queens
Now we can't completely do away with this self-similar notion, because it's an idea which is helpful in studying the dimension of figure. In fact, this iterative way of looking at figures is what birthed fractals in the first place. Take a look at how self-similarity works on the "peaceful queens" problem..

Suppose you have a chessboard which is k squares by k squares. How many Queens can you place on the chessboard without putting them in position of attacking each other, and how must they be arranged? This can be a very complicated problem, but let's look at a 5x5 board. It turns out that the best way to arrange the queens is like the figure to the left.

Now it turns out that the best way to arrange queens for a 25x25 board looks like this:

See the self-similarity? It has been proven that you can take a 125x125 board and use the same placement again to get the most peaceful queens on the board. Interesting, huh? It turns out that this is the underlying principle which makes this work is how fractals work, all involving self-similarity. (This has been taken from the introduction of "Fractals, Chaos, Power Laws" by Manfred Schroeder, they got the idea from an article by Clark and Shisha

Bottom line: fractals often exhibit self-similarity, and the notion is helpful when studying fractal geometry, but it is certainly not their defining characteristic.