In real life chemistry, 'principle' means the active chemical in a compound, the one that makes it work. It's complicated.
This is some guesswork on my part, but from what we can gather in the first two books:
Alchemy involves mixing chemicals, or "principles" and then performing something like a sympathetic binding on them. The binding nullifies the normal chemical properties of the materials and replaces them with the magical properties you'll get from using the potion.
However, the process isn't perfect. Some of the principles remain unbound and retain their normal chemical properties, which can impact the person consuming the potion. So you take the time to do some chemical processes that will neutralize the unbound principles, or in some cases, precipitate the undesirable elements so you can filter them out. And then you have to make sure the neutralizing agents themselves won't harm the consumer, and so on, so you might have to go through multiple purification rounds before it's actually safe to drink.
The amount of unbound principle remaining in your finished potion depends on a lot of things, like the Alar of the alchemist and the materials involved. For example, heavy metal compounds might be particularly resistant to binding, and an alchemist with a weak alar will get less potion and more unbound principles than an alchemist with a powerful alar.
So if you're trying to quickly brew up a potion to attack somebody you hate, you probably don't bother with purification, and the potion has a lot of unbound principles that can cause their own brand of havoc for the consumer.