A fibered category (or Grothendieck fibration) over a base category B is a functor p: E → B such that for every morphism f: b → b' in B and object e' in E with p(e') = b', there exists a cartesian morphism (or cartesian lifting) φ: e → e' with p(φ) = f, satisfying a universal property. Intuitively, E is a "family of categories parametrized by B": the fiber E_b = p^{-1}(b) is the category sitting over each object b, and cartesian morphisms provide canonical ways to pull back along morphisms in B. Fibered categories formalize the notion of "varying algebraic/geometric structure over a base" and are central to descent theory, stacks, and the Grothendieck construction.
Consider the codomain fibration cod: Arr(C) → C sending each arrow f: A → B to its codomain B. The fiber over B is the slice category C/B. A cartesian morphism is a pullback square. Verify the universal property of cartesian liftings in this example. Then consider the category of vector bundles over a base space, where the fiber functor sends each bundle to its base space—pullback of bundles provides the cartesian liftings.
From your study of comma categories and functor categories, you know how to form slice categories C/b and how to assemble categories of functors into structured wholes. Fibered categories answer a subtler organizational question: how do you talk about a *family* of categories varying over a base, where morphisms in the base tell you how to transport objects between fibers? The motivating example is geometric: over each topological space (or scheme) B, you have a category of vector bundles — small changes in the base space induce pullback operations on bundles, and those pullbacks are the "transport maps" the fibration formalizes.
Formally, a Grothendieck fibration is a functor p: E → B such that for every morphism f: b' → b in B and every object e in E with p(e) = b, there exists a cartesian morphism φ: e' → e in E with p(φ) = f, satisfying a universal lifting property. The universal property says: any morphism ψ: e'' → e in E with p(ψ) factoring through f in B factors uniquely through φ. Concretely, φ is the "best" way to lift f to E above the target e — it is canonical once the target and the base morphism are fixed. The object e' = f*(e) is the pullback of e along f, and this is exactly a pullback in the geometric sense when E is the category of vector bundles.
The fiber E_b over an object b ∈ B is the subcategory of E consisting of objects e with p(e) = b and morphisms φ with p(φ) = id_b. In the vector bundle example, E_b is the category of vector bundles on the single space b. A morphism f: b' → b in B induces a pullback functor f*: E_b → E_{b'}, defined (up to canonical isomorphism) by the cartesian liftings. This is where your comma-category intuition applies: the comma category b'/p in E records all ways to map into a fixed fiber. The codomain fibration cod: Arr(C) → C, sending each arrow g: A → B to its codomain B, has fiber (Arr(C))_b = C/b — precisely the slice category over b. The cartesian morphisms are exactly the pullback squares in C.
A cleavage is a choice, for each (f, e) pair, of a specific cartesian lifting φ_{f,e}. Different cleavages give equivalent total categories, but choosing one converts the fibration into a strict assignment b ↦ E_b and f ↦ f* that almost forms a pseudofunctor B^{op} → Cat — the composition f* ∘ g* need not equal (g ∘ f)* on the nose, only up to coherent isomorphism. A split fibration is one where the cleavage is strictly functorial: (g ∘ f)* = f* ∘ g* strictly. Most natural fibrations are not split, but every fibration is equivalent to a split one.
The deeper significance is the Grothendieck construction: there is an equivalence (more precisely, a 2-categorical equivalence) between fibrations p: E → B and pseudofunctors F: B^{op} → Cat. Given a fibration, you extract F by b ↦ E_b and f ↦ f*. Given a pseudofunctor, you build the total category E with objects (b, x) where x ∈ F(b), and morphisms (b', x') → (b, x) given by pairs (f: b' → b, α: x' → F(f)(x)). This is precisely why fibered categories are central to descent theory: a presheaf F: B^{op} → Set (or Cat) descends along a covering if and only if the corresponding fibration satisfies a gluing condition. Stacks are fibered categories over sites that satisfy the descent conditions, and the entire language of algebraic geometry over varying base schemes is organized through this formalism.