I know of at least three equivalent formulations of QM:
The "normal/standard" one, dealing with Hilbert spaces and state vectors.
The Feynman path-integral formulation.
The Wigner-Weyl phase space formulation.
My question is: what is the usual name given to the first, "normal" formulation that everyone learns as an undergraduate?
The normal interpretation is the Copenhagen one.
A common formalism is the position representation of the nonrelativistic Schrödinger wave equation in the Schrödinger picture (with $L^2$ as your Hilbert space for bound states, $\frac{\hbar}{i}\nabla$ as the momentum operator, $x$ as the position operator, operators constant, states being wavefunctions changing in time).
Variations would be to include:
1) Making wave functions constant, operators changing in time, this is the Heisenberg picture.
2) Making the wavefunction be a function of momentum instead of position so the operator for momentum is $p$ and position becomes a differential operator, this is the momentum representation.
3) Using the Pauli wave equation instead of the Schrödinger wave equation, so you can handle spin.
4) Enlarging your Hilbert space to beyond $L^2$ to include unbound states.
5) Adding extra terms for relativistic corrections.
6) Fock spaces, second quantization, density matrices, states as positive linear functionals on operators, weak measurements, decoherence analysis, etc.
The Hilbert space formulation, sometimes with the explanatory add-on: "where observables are represented by linear operators acting on Hilbert space". Both the wave and matrix sub-formulations are basically shadow-double formalisms for the very same structure.
You are probably referring to the Dirac picture, where vectors are represented by ket vectors, and linear functionals are given by "bra vectors". This picture is equivalent to both the Schrödinger and the Heisenberg picture. See this answer for more details.
