The two synonyms "fractal" and "fractal object" come from the Latin "fractus" meaning irregular or discontinuous. The objects these words refer to and the objects fractal geometry refers to have very irregular shapes, therefor more sophisticated objects like the ones we have around us can be described and analyzed. Euclidean or classical geometry deals only with regular shapes, shapes that are unlikely to be found in nature. This is the first application of fractal geometry: to analyze objects Euclidean geometry can not analyze. Fractal geometry can analyze any object no mater how irregular it may be identifying it with a fractal object.
One of the main characteristics of a fractal object is its fractal dimension (D) that measures the objects' degree of irregularity and discontinuity. A very important fact about a fractal dimension is that unlike regular dimensions (which can only have integer values) it can be a simple fraction like 1/3 3/7 or even an irrational value like lg3/lg7»0,5645... or p. Classical geometry teaches us that a point or a multitude of points is zero-dimensional, a straight line is one-dimensional a surface or a plane is two-dimensional and a cube for instance is three-dimensional. We will consider that a very irregular plane curve has a fractal dimension between 1 and 2, a leaf shaped surface with a lot of irregularities has a dimension between 2 and three and a trace of dust has a dimension between 0 and 1. An objects dimension may vary depending on the resolution we look at the object with. For example lets consider a ball of wires 10-cm in diameter and with 1-mm thick wires. If we look at it at a 1-m scale we see it as a dot, therefor it is zero-dimensional. If we zoom in to a 10-cm scale we will see it as a three-dimensional sphere. If look at it at a 10-mm scale we will see a lot of wires therefor a one-dimensional shape. If we zoom in more the wires will look like columns so the image is three-dimensional. As we decrease the scale the wires are going to split into smaller threads so the image is now one-dimensional again. If look at a molecular scale we will see a multitude of atoms so a zero-dimensional image. If we look at it at an intermediary scale we might see some transition areas that don't look like a zero, one, two or three-dimensional object. These transit areas will be regarded as fractal objects having a fractal dimension.
Still some fractal objects may have an integer dimension like 1 or 2 but they don't look like a plane or a line.
Some mathematical procedures of dealing with chaotic shapes have existed for some time but weren't given much attention because they were considered unusable and complicated. As soon as applications of fractal objects became more common more attention was drawn to this branch of science and a new way of looking at things took shape. Between the uncontrollable domain of chaos and the excessive order of Euclide
The disciplines of complexity, after a period in which even scientists treated them with uncertainty, are very popular not because of "magic words" like "chaos", "calamity", "fractal", but due to orientation towards biological related fields, natural shapes, evolving dynamic systems which practically can not be described using classic physics.
Chaos Theory got its name thanks to the American mathematician James Yorke, when he attempted to name the theoretical approaches of irregular (chaotic) shapes.
This name has a slight semantic ambiguity because we are not talking about chaos in it's true meaning, meaning dynamic systems that in their behaviour defy any attempt of rational covering, but systems that have a certain structural stability. In Chaos Theory we won't find any ideal reductionist mobiles like ideal pulley, mathematical pendulum, ideal gas etc. but openings towards understanding unordered processes and irregular shapes very often found in nature. The irregularities of the course of a river which seams the same but still is different every time (Heraclit associated it very well in ponta rhei), the trajectory of a falling leaf, the trace a flying jet leaves behind, turbulence etc. are such examples. We must understand, at least up to a point, that in the classic paradigm there was no room for such things because the mathematical device needed to describe such phenomenons is very sophisticated, oftenly requiring the use of computers in order to solve them. It is said about Heisenberg that he promised that when he would cross over to the world of the shadows he would ask God two things: "Why do relativity and turbulence exist?". "I'm sure he could answer the first question" Heisenberg would have added.