For example; what is a "position bra" (under-wired ? ), or a "state ket", and what is the significance of < | and | >.

It depends on the level of the answer [

]

A ket is a state vector, a bra is a linear functional which associate a complex number to a ket.

But I suspect you would prefer something simpler.

In qm you use "wavefunctions" which are complex valued functions of x, y, z, t and which describe the qm system (for ex. a particle): the wavefunction codify the informations about the state of the system. But you need to make the scalar product of two wavefunctions f(x,y,z,t) and g(x,y,z,t); this is represented with the symbol: <f,g> and it is usually computed as:

<f,g> = Integral[-oo,+oo] f*(x,y,z,t) g(x,y,z,t) dxdydz

where f* indicates the complex coniugated of f.

It turns out that the integral is much better viewed as a function which associate a complex number (the result of the integral) to the function g, given the function f, so it is a "functional", that is something which take a function and returns a scalar (instead a normal function takes a scalar and returns a scalar).

Dirac invented this notation: the scalar product can be viewed as the functional <f| applied to the function |g> and becomes, simply: <f|g>. Since it was formerly represented with a "bracket": (f,g), he said that the firs part is the "bra" and the second the "ket" [

] and so the name.