Peano curve is a name give to any fractal whose fractal dimension is equal to 2. However, the name for this type of fractals comes from the name of the original Peano Curve. The original Peano Curve is a base-motif fractal which uses a line segment for the base and the following motif:
Often, a square is used for the base instead of a line segment. To generate the Peano Curve, you start with a line segment and substitute it with the motif. You then take every one of the 9 line segments in the figure and substitute it with the motif again. At the end, you get a square:
The fractal dimension of all Peano curves is 2 by definition. You can find it in the above fractal using the similarity method. In the motif, there are 9 identical line segments, the size of each of which is 1/3 of the original line segment. The fractal dimension is, thus, log 9 / log 3, which is equal to 2.
After every iteration, the Peano Curve becomes 3 times longer. After an infinite number of iterations, it will become infinitely long.