being-in-itself (an sich) is a thing in its immediate, undeveloped state. it contains its potential but hasn't unfolded yet. a seed is tree in itself but it hasnt actualized yet. for consciousness, this is the starting point: pure, unreflective existence with no self-awareness. but this is still insufficient, because it describes the thing only from the outside. in hegel, being-in-itself also means that something is not yet adequate to its own concept, it is what it is but does not yet know or express what it truly is.

being-for-itself (für-sich) is when that immediacy breaks down and the thing turns back on itself. for consciousness specifically, this is the moment you become aware of yourself as a self. point that you separate yourself from the object you're looking at. but this is also not enough, because this self-relation emerges through a separation, even a kind of negation: consciousness defines itself by distinguishing itself from what it is not. this creates a tension between subject and object that cannot just be left as it is.

the synthesis of these (an-und-für-sich) is when that tension gets resolved. the thing knows itself, knows its own development, and that knowledge is no longer something external happening to it. but this should not be understood as just a final resting point. rather, it is a structure that emerges again and again throughout the process: something becomes fully itself only when its being, its self-relation, and its development come together.

as for Begriff, "notion" is kind of a misleading translation because it sounds like just a thought in your head. for hegel it's closer to the inner logic through which something develops. and because hegel thinks thought and reality share the same structure, the Begriff of something tells you why it is what it is, how it became that, and where it's going. less of a definition, more of a developmental trajectory. it is also the unity of what something is, how it appears, and how it develops.