Background Knowledge is the abstract collected sum of a character's understanding of the underlying theories, best practices, commonalities, etc. of one of the five disciplines of magic. No witch knows everything, and even those more skilled in certain disciplines have areas of greater speciality and areas where they just know the basic facts. A witch with a greater degree of base background knowledge is considered more powerful in that discipline, though this is on average, as they have a greater understanding of the underlying behaviours and interactions in their chosen discipline. Furthermore, when attempting to recognise a clearly magical effect, or perform research in areas related to a discipline then the greater penumbra surrounding their background knowledge is going to be very helpful. That being said it is unusual for a witch to focus their studies exclusively on one discipline, as the comparison and contrasting of different disciplines often helps highlight the distinctions within a witch's discipline of choice.
When attempting to cast a spell in your turnsheet, you should refer to your character's background when describing the effect that you are attempting to create. This works in the same way that you would refer to your skills when attempting to make mundane changes; due to the nature of magic, the effect you have caused isn't always immediately obvious but making it explicitly clear what you are trying to achieve is very helpful for the GMs. Here, intention is quite literally magic.
You may continue to take advancement points in Background Knowledge in play, but may not exceed 10 points spent, total in a single discipline.
When taking another point of Background Knowledge, as well as getting a general increase in your knowledge and power, you gain a specialisation in a certain area of the discipline. This specialisation - also known as a solution - takes the form of a verb that is thematically in keeping with the school as a whole. If you can describe the effect you are trying to achieve with your spell using this verb it is more likely to be successful and effective. It is not necessary to explicitly state this in your turnsheet to gain this bonus, but putting a comment in noting this is more likely to get it recognised by the GMs.
Examples are given below - these are not an exhaustive list, and you are free to suggest other specialisations subject to GM approval.
At character generation you must take 5 advances of Background Knowledge. As well as providing your character's base level of understanding, they also guide the Skill picks you can make - one advancement in each for each advance in Background Knowledge (in the same disciplines).
The study of beauty, of meaning through perception, of art, of the senses.
The study of the body, of anatomy, of movement through space.
The study of morality, of philosophy and of belief.
The study of language, of words, of semantics, and the creation of written works.
The study of numbers, quanitities, spaces and the application thereof.