Notation used in quantum mechanics.

Ket is just a vector.

Bra is just the projection operator on a given vector, also known as... dot product ;-) Like the dot-product, it operates on a vector and returns a number.

This notation is widely used in quantum mechanics because calculating the probability of a given quantum state is the same as calculating the projection on an eigenvalue basis vector as explained at: Section 1.2.1.1.4.5.3. "Mathematical formulation of quantum mechanics".