What is meant by protons having positive charge? I am really confused in physics with my course chapter "Electrostatics".
So I used google to search for my answers but I'm not satisfied with them.
What is the meaning of a proton having positive charge? Does that mean protons are tiny particles having more protons than electrons? Is that why protons have positive charge?
My professor wrote down this definition of charge:

There is no particular definition of charge but its the property by which electrons and protons attract each other.

 A: When physicists created the sources of electricity, they called one pole positive and the other negative. They decided that electric charges move from the positive pole to the negative pole. Then it turned out (later on) that the electrons move in the opposite direction, so the electrons are charged negatively, according to the previous convention. The protons are the charged positively, and in electrolytes they indeed move from the positive pole to the negative one. Historically it was so, and the main thing to keep in mind is that the electron and the proton have charges of the opposite sign. 
A: A simple (but essential, mathematical) answer might be as follows. Dirac by deriving his equation from the relation:
$$E=±(m_0^2c^4-p^2c^2)^{1/2}$$
(which formally defines the relativistic expression for the energy)
showed that the minus sign in this relation implies the existence of anti-particle with an opposite charge sign (than the particle). On this basis, when we have electrons with negative charges (we should have positrons with positive charges), when we have anti-protons with negative charge (then we also have protons with positive charges) and so on. So, the existence of two different particles with opposite charge signs (such as electrons and protons) is quite natural and logical in mathematical point of view!
A: And to add to @Orion's perfect answer:
A) charge is a fundamental property of an elementary particle, it may be +1, -1, or 0, in natural units with the electron denoted as -1, or -e, except for quarks which come as -1/3's or +2/3's. Charge is fundamental in the same way that mass is, each elementary particle has some value of each (and also of spin). 
B) the quarks that Orion mentions for the proton are two up (u) quarks (at 2/3 each, and one down (d) quark at -1/3, making the total the 1 value of positive charge, or +e. 
C) there are 6 quark types, including the u and d, and they make up the protons, neutrons, antiprotons, mesons etc. They are held together by gluons, which have no charge. Gluons and quarks, and quarks with each other, interact through the strong nuclear force. Quarks also have electromagnetic interactions with each other since they have charge. The proton, made up of 3 quarks, is an extremely stable configuration. That's why the universe is full of them. They are in every atom. 
D) Electrons are fundamental particles, not composed of anything more basic, at least up to the energies we've been able to explore. 
E) There are other fundamental particles, see the Wikipedia article about quarks and a table of the fundamental particles of the Stabdard Model, the current understanding of fundamental particles in physics. See at https://en.m.wikipedia.org/wiki/Quark
F) The charge has one and only one effect: it is responsible for the electromagnetic force, with electrostatic meaning charges at rest and thus no magnetic field, only electric. You'll get more in your course or later. 
