Maybe I have overseen it in the Tutorial, but how exactly do I get a Potent Aspect? What governs the creation of this?
There's more than one way
The most common is definitely the Forest Giant's Fertility Boost ability. If you place an Aspect on a square while the fertility boost is active on that square, there is a high chance it will be a Potent Aspect.
The Water Giant's Moonsoon ability works the same way, however it needs to be unlocked before you can use it, and it covers a range of squares instead of just one.
You can also find naturally boosted spaces at the start of the game. Just look around for the faint up-arrows above a space. When you click on the space, you'll see a green icon that tells you the space has Level 1 Fertility on it for 10 minutes.
And there is a chance to get a Potent aspect anytime you place one. The starting chance is 25%, however if you have a higher level aspect, you have a better chance at getting a Potent or better aspect.
You have to fertilize the ground with your forest or water giant before you cast an aspect. That enhances the aspect and makes it a potent aspect, or, if you have the ability, it makes it a greater or sublime aspect.