Projective, or POVM, and why the difference is the whole security argument behind QKD
Postulate 3 (topic 01) stated the general rule for measurement. Two special cases of it come up constantly, and they answer different questions.
Projective measurement is the textbook case: measuring "along" some observable, getting one of its eigenvalues as the outcome, and the state collapsing onto the corresponding eigenspace. It's precise, but it forces the state onto one of a fixed set of directions, which is sometimes more than you actually need.
POVMs (positive operator-valued measures) generalise this: instead of a fixed orthogonal basis, you can measure with a set of operators that overlap, don't need to be orthogonal, and don't need to correspond to a clean post-measurement state. They're the right tool whenever the question is only "what's the probability of each outcome," not "what state does the system end up in."
Both obey the same hard limit: you cannot reliably distinguish two non-orthogonal states, by any measurement quantum mechanics permits. That is not a limitation of current technology. It follows directly from the postulates, and quantum key distribution turns it into a security guarantee: an eavesdropper measuring a qubit prepared in one of two non-orthogonal bases cannot both learn its value and leave it undisturbed.
Projective measurement is specified by an observable: a Hermitian operator M with spectral decomposition M = Σm m Pm, where {Pm} are orthogonal projectors (PmPm'=0 for m≠m', Σ Pm=I). Outcome m occurs with probability ⟨ψ|Pm|ψ⟩, and the post-measurement state is Pm|ψ⟩ normalised.
POVM measurement is specified by a set of positive operators {Em} with Σm Em = I. No orthogonality required, and Em need not even be a projector. Outcome m occurs with probability ⟨ψ|Em|ψ⟩. Every projective measurement is a POVM with Em=Pm; the converse doesn't hold, which is exactly why POVMs are strictly more general.
Distinguishability. Two pure states |ψ⟩ and |φ⟩ can be perfectly distinguished by some measurement if and only if ⟨ψ|φ⟩ = 0. If ⟨ψ|φ⟩ ≠ 0, no measurement (projective or POVM) can identify which one you were handed with zero error probability; the best achievable success probability for a single copy is bounded by the Helstrom bound, ½(1 + √(1−|⟨ψ|φ⟩|²)), which is strictly less than 1 whenever the states overlap.
Quantum key distribution BB84 encodes a bit in one of two non-orthogonal bases on purpose. An eavesdropper who measures in the wrong basis necessarily disturbs the state: the distinguishability limit above, made into a security protocol. This is the topic post-quantum cryptography is most often confused with: PQC is classical mathematics designed to survive a quantum computer; QKD is physics that uses exactly this measurement limit to detect an eavesdropper. They solve the same problem (safe key exchange) by entirely different means; the full comparison, including why every major cybersecurity agency still recommends PQC over QKD for this, is on that page.
Check yourself
Two quantum states have a non-zero overlap (the inner product of psi and phi is not 0). What can any measurement do to tell them apart from a single copy?
Two pure states can be told apart with certainty if and only if they are orthogonal. Otherwise the best success probability is bounded by the Helstrom bound, which is below 1. That is the limit quantum key distribution turns into a security guarantee.