Self-similarity simply means that the parts of the object are similar to the whole object. A fractal is composed of several smaller versions of itself. Self-similarity is an important characteristic for the creation of realistic fractals. Fractals also have a dimension associated with them. It is in no way similar to Euclidean dimension.
Fractal dimension is a measure of the degree of roughness or fragmentation of an object.
The dimension of a fractal can be a fraction. The greater a figure's dimension, the greater it's complexity. All fractals have some pattern that repeats itself infinitely. The creation of a fractal results from the iteration of the steps that create the repetitive pattern within the object. Generator iteration creates base-motif fractals.