inner product
The inner product, written ⟨φ|ψ⟩, is the single complex number you get by pairing a bra with a ket. It is the quantum version of the dot product you may know from ordinary vectors, and it measures how much two states overlap — how alike they are. If the number is large in size the states are similar; if it is zero the states are orthogonal, meaning they share nothing and are perfectly distinguishable.
Unlike the dot product of everyday arrows, the quantum inner product can be complex, and the order matters: swapping the two states gives you the complex conjugate, so ⟨φ|ψ⟩ and ⟨ψ|φ⟩ are mirror images of one another. Pairing a state with itself, ⟨ψ|ψ⟩, always gives a real number that is zero or positive; its square root is the length, or norm, of the state, and we usually scale states so this length equals one.
This one operation quietly powers almost everything quantum. Probabilities come from the squared size of overlaps, by the Born rule; expectation values are inner products with an operator sandwiched in between; and asking whether two states can be told apart in a measurement is really asking about their inner product. Geometry — lengths and angles between state vectors — is the language in which quantum predictions are phrased.
The inner product is conjugate-symmetric and gives each state a real, non-negative length.
An overlap ⟨φ|ψ⟩ is not itself a probability — it is a complex amplitude. Only its squared size |⟨φ|ψ⟩|^2, via the Born rule, gives the probability of finding state ψ to be φ in a measurement.