Metamorphosis.
noun: a change of the form or nature of a thing or person into a completely different one, by natural or supernatural means. [1]

From this graph, you can see that this concept has been around and in discussion at a fairly constant rate. The idea that an one object can turn into a very different one, is fascinating for various reasons. How does this happen? Or more importantly, why?

To answer these questions, we need to observe it in action.
There is geological metamorphosis, where changes occur in minerals or geologic textures. It takes place due to temperature, pressure, and chemically active fluids. There is no underlying 'why', here, except that rocks that are placed in certain conditions behave in certain ways (according to the laws of physics and chemistry).
Then, there is biological metamorphosis, a completely different beast (sorry, couldn't resist). This process involves significant change in structure due cell growth and differentiation after birth/hatching. This process is so successful, that the majority of all insects (and therefore the majority of all animals) undergo some form of metamorphosis.
Let's look at this in a little more detail, in the case of the Monarch butterfly (Danaus plexippus). The monarch butterfly is part of the homometabola subclass of insects that undergo complete metamorphosis, passing through a distinct pupal stage.

Danaus plexippus start out their life as a caterpillar, hatched from a tiny egg. For around two weeks, they roam around their host plant, eating all the leaves they can find. This allows them to grow more than twenty-fold, from a length of ~ 2mm, to ~ 5 cm.
In the image below, the one on the right has just hatched, and will grow in size for around 13 days till it becomes the size of the one on the left.

https://www.math.auckland.ac.nz/~hafner/monarch/

But what comes next is what we're interested in.
The caterpillar finds a nice spot, attaches itself to a branch, hangs underneath, and forms a pupa (also called chrysalis).
This beautiful video shows the pupation process:

And after another 2 weeks in the pupa, they emerge, as the beautiful, photogenic, Monarch Butterfy.

The Monarch Butterfly  (http://en.wikipedia.org/wiki/Monarch_butterfly) — The Monarch Butterfly (http://en.wikipedia.org/wiki/Monarch_butterfly)

But, wait.

What just happened?

How did a creature go from being:

Stop reading for a moment, and think about it. How do you think this transformation happens?

Thought about it? Ok.
Scientists a few hundred years ago wanted to know too, so they took a pupa, and cut it open.

And what did they find?
Goo.
Pale yellow goo.
Are you surprised?
There was no caterpillar, no butterfly, just a snot colored liquid. No head, no antennae, no legs. Where did the caterpillar go?

It seems like once the caterpillar gets into it's chrysalis, it releases enzymes that dissolve it's tissues into individual cells. Some of these cells rupture, and when they do, their contents (the proteins, cytoplasm) all spill out.

When early scientists first saw this, they came to the conclusion that the caterpillar entered the pupa, effectively died, and out of its remains was reborn as a butterfly. This explanation was used for a long time as an example by the Church to demonstrate how the human body might die, and attain a more beautiful form in heaven.

This idea was squashed completely in two different ways. The first is an ingenious experiment run by scientists in the UK. Here's what they did. They took a bunch of Manduca sexta (also called the tobacco hornworm, a species of moth) while they were in caterpillar form, sprayed them with an odor, and then shocked them (with electricity). After they did this often enough, the caterpillars learned to hate the odor, and everytime they smelt it, they tried to move away. Now, they let those caterpillars pupate. They spent two weeks in the pupa, and emerged as moths. Now, the scientists gassed them again...... they hated it! Ordinary unconditioned moths did not have any specific reaction to the odor, but these moths hated it, and kept trying to avoid it. [2]

Why? What does this mean?
It means that the memory made it through the goo.
What the caterpillar learned, the moth remembered. This conclusively proved that the caterpillar didn't die in the pupa, it simply transformed.

Another experimenter that subverted the "dying" hypothesis was a Dutch microscopist called Jan Swammerdam. He routinely performed dissections of dragonflies and mayflies and examined them under a microscope. On dissecting the caterpillar form of those insects he was able to show undeveloped structures of the adult. So on peeling back the skin of the caterpillar, you found tiny, thin wings, antennae and legs of the future adult, even before pupation began.
This proved, beyond a doubt, that the insect before pupation and after was the same organism, with shared organs.

Today, we call these immature little adult structures imaginal discs. The goo in the pupa actually contains lots of these bunches of cells, which use the nutrients spilled out of the other cells to fuel growth of those parts of the adult body.

In Part 2, we explore how metamorphosis is actually useful, why it was evolutionarity selected for, and can we see what goes on inside the chrysalis?

Sources and Further reading :

[1] http://en.wikipedia.org/wiki/Metamorphosis
http://en.wikipedia.org/wiki/Endopterygota
http://www.scientificamerican.com/article/caterpillar-butterfly-metamorphosis-explainer/
http://www.scientificamerican.com/article/insect-metamorphosis-evolution/
http://www.radiolab.org/story/goo-and-you/

[2] http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0001736