The emergence of a moth or butterfly from its pupa or cocoon is hard to predict. It depends on weather, time of year, genetic variations, food availability to the larva, and whether the animal inside has died or not. Some adults emerge in 2 weeks, some in 2 years, so waiting for the adult to emerge may require some patience. (Note: caterpillars 'hatch' from eggs, but adult Lepidoptera 'emerge' from their pupae)

A chrysalis must not get too wet (or it gets attacked by mould) or too dry (or the animal inside dies of dessication). It should be disturbed as little as possible as pupae bruise easily.

Provide some twigs for the new adult, when it does emerge, so that it can hang upside down, as most Lepidoptera need to do this for their wings to expand properly.

Chrysalis and girdle of Papilio aegeus, PAPILIONIDAE

Your patience may also go unrewarded: a high proportion of Caterpillars get infected with parasitoids, and these tend to make the Caterpillar upset and wander about instead of hiding on its food plant: and so these are the ones that are most often found. So if a bunch of flies or wasps come out the pupa, do not be too surprised.

parasitoids on a Cryptoptila immersana Caterpillar, TORTRICIDAE

Once you have an adult, make sure it does not beat itself to a frenzy (butterflies and moths basically do not like being kept in jamjars): a spell in the refrigerator (NOT the freezer) is a good way of putting them to sleep. Then you may finally be able to identify it from pictures in these web pages, or in books, or with the help of your local museum.