Is an orbital made of two spinorbitals? I am a chemistry student,i dont know much about quantum chemistry, but i've studied, that an orbital is represented by one electron wave function. 
And we always fill one orbital with two electrons with different spin.
Does it mean that for example 1s orbital is made of 2 spinorbitals, one for an electron with spin up, one for an electron with spin down?
Does it mean than 1s orbital is made of two wave functions, considering it has two electrons?
Thank you!
 A: Also Yes. Electrons want to be in the lowest energy state possible--the 1S state. So, if you have a bare nucleus and add an electron, it goes to the 1S state--for which the "1" means it's the lowest energy radial wavefunction, and the "S" means it is spherically symmetric (L=0).
Add another electron, it also goes to the 1S state. Now since there are 2 electrons in the same spatial state, there spin state must be antisymmetric. It's common to say "they have opposite spins", but that is colloquial. Neither is in a definite spin state, rather there total spin wavefunction is S=0.
Continuing this process builds the Periodic Table, though there are subtleties regarding the order in which the shells are filled, and other things that affect the energy of a state: nuclear magnetic momentum, spin-orbit effects, finite nuclear size, relativistic corrections, to a name a few.
If you add a 3rd (and 4th) electron, the lowest energy state is filled--the Pauli Exclusion Principle blocks it from falling into the 1S state. It goes to the 2S states, which has a different radial wavefunction from the 1S state.
Note that if had added a $\mu^-$ (muon), it would fall into the 1S state because it is not an electron, so the Pauli Exclusion Principle does not apply.
When you get to the 5th-10th electrons, the angular part of the wave function has angular momentum (2P). This shell holds 6 electrons because there are 3 different orbital angular momentum states with L=1.
A: Short answer, Yes!
Each electron has its own wave function which describes the particle. The quantization problem of the Schrodinger wave equation in spherical coordinates gives us the solutions showing different orbitals in the atom. More of this can be found at this link.
Later when the Zeeman effect was observed, the exact doubling of lines couldn't be explained. The same effect was observed in the Stern-Gerlach experiment. Spin, along with the other quantum numbers, describe the distribution of electrons around the nucleus. In the end, Pauli's exclusion principle makes it clear that each electron has its own unique set of quantum numbers, which are nothing but parameters which characterise the wave function. And different quantum numbers mean different wave equations.
A: If you like to be more precise, no. You will recognize that the terms of s-,p-,d- ... -orbitals are based on the solutions of the Schrödinger equation for the hydrogen atom, i.e. an electron moving in a spherical symmetric potential. Spin is not included at all in this type of problem; if you have a look at the eigenfunctions, i.e. the wave functions refering to the well known orbitals, you will not find a quantum number related to spin, which explicitly determines the nature of the wave function. From a quantitative point of view you need to extend your theory in order to describe spin and spin-orbitals, respectively. Therefore, it would be necessary to study relativistic quantum mechanics based on the Dirac equation.
In standard chemistry, one usually does not include the spin degree of freedom on the level of spin orbitals but it will appear as additional quantum number, which characterizes the orbital and often only appears as additional index. Therefore, I would not recommend to argue a 1s orbital with two electrons is equivalent to two spin orbitals refering to the different electrons sitting in a 1s orbital. You have here two different perspectives refering to two different theoretical approaches based on the Schrödinger equation or Dirac equation, respectively. 
