That's an important point. So let's clarify it once and forever:
Picture: In the abstract formulation of quantum theory in terms of selfadjoint operators, representing observables on a Hilbert space and the Hilbert-space vectors, representing states you can shuffle the time dependence between these operators and states pretty arbitrary. The specific way you do this "split of the time dependence" is called the choice of the picture of time evolution. Which one to use is just a matter of convenience. The physical outcome of your calculation, i.e., probabilities, expectation values of operators for a given state are independent of the choice of the picture.
The Schrödinger picture is the one, where the entire time dependence is put on the state vectors and thus the operators representing observables are time independent. In the Heisenberg picture it's the opposite. The general idea of the picture was discovered by Dirac, and usually applied as the "interaction picture" in time-dependent perturbation theory.
Representation: A representation is the choice of which basis you use to formulate quantum theory in terms of matrix or wave mechanics (or a mixed form, depending on the basis you choose). A basis is a (generalized) set of orthonormal common eigenvectors of a complete set of compatible operators. For a spinless non-relativistic particle common choices for wave mechanics is the position representation, where the complete set of compatible operators are the three components of the position vector operators or the momentum representation (three components of the momentum vector). A mixed representation in terms of a wave-matrix mechanics is given, e.g., by choosing ##|\vec{x}|=r##, ##\vec{L}^2##, and ##L_z## as a complete set of compatible observables. Then you have a mixed representation in terms of wave functions ##\psi_{lm}(r)## with ##r \in \mathbb{R}_{>0}##, ##l \in \{0,1,\ldots \}##, ##m \in \{-l,-l+1,\ldots,l-1,l \}##. Finally an example for matrix mechanics is the choice of phonon eigenstates of a 3D harmonic oscillator, because there you have a basis given in terms of three occupation numbers for the modes of the oscillator.
The time evolution of the corresponding concrete realization of Hilbert space in terms of wave functions, infinitely-dimensional matrices, or a mixture there of is independent of the choice of the picture. Because the time dependence of the eigenvectors of observables, appearing in the left argument of the generalized scalar product that defines the "vector components" ("wave functions") of your representation, and the time dependence of the state ket, appearing in the right argument of this scalar product always combine to the unique Schrödinger equation, written in this representation, e.g., in the position representation
$$\mathrm{i} \hbar \partial_t \psi(t,\vec{x}) = \hat{H} \psi(t,\vec{x})=\left [-\frac{\hbar^2}{2m} \vec{\nabla}^2 + V(\vec{x}) \right ] \psi(t,\vec{x}),$$
where
$$\psi(t,\vec{x})=\langle \vec{x},t |\psi,t \rangle.$$
Of course, you must use the same picture of time evolution for both the eigenvectors of the operators representing the complete set of compatible operators and the state ket in this scalar product.
Here ##\hat{H}## is the Hamiltonian, which by definition is the generator of the physical time evolution. In the abstract formalism, it is very illuminating to introduce a covariant time derivative, which is however rarely done in textbooks (I only know a German one, where this is done: E. Fick, Einführung in die Grundlagen der Quantenmechanik). It consists of the postulate that if ##\hat{O}## is representing an arbitrary observable, then
$$\mathrm{D}_t \hat{O} = \frac{1}{\mathrm{i} \hbar} [\hat{O},\hat{H}]$$
represents the time derivative of this observable. This leads to a picture-independent formulation of quantum theory in the abstract Hilbert-space formalism.
