"Photosystem" and "Reaction center" are not synonyms.
According to Lodish (2000):
The absorption of light energy and its conversion into chemical energy occurs in multiprotein complexes, called photosystems, located in the thylakoid membrane. A photosystem has two closely linked components, an antenna containing light-absorbing pigments and a reaction center comprising a complex of proteins and two chlorophyll a molecules. (emphasis mine)
And according to Alberts (2002):
Multiprotein complexes called photosystems catalyze the conversion of the light energy captured in excited chlorophyll molecules to useful forms. A photosystem consists of two closely linked components: an antenna complex, consisting of a large set of pigment molecules that capture light energy and feed it to the reaction center; and a photochemical reaction center, consisting of a complex of proteins and chlorophyll molecules that enable light energy to be converted into chemical energy. (emphasis mine)
Therefore, a photosystem is composed by:
- The antenna complex;
- The reaction center.
Here is a figure from Alberts (2002):
Figure: The antenna complex is a collector of light energy in the form of excited electrons. The energy of the excited electrons is funneled, through a series of resonance energy transfers, to a special pair of chlorophyll molecules in the photochemical reaction center. The reaction center then produces a high-energy electron that can be passed rapidly to the electron-transport chain in the thylakoid membrane, via a quinone.
Thus, in conclusion, the reaction center is part of the photosystem.
PS: Both quoted texts answer your second question ("And if the reaction center is only a part of these PSI and II complexes, what does it contain? And what does it do?").