Disclaimer: All mathematical expressions and the definition of the appearing symbols are to be considered sketches.
Some books say that because of this field the matter has the properties of mass but when I see arxiv I think it's something different.
There is effectively some difference between the Higgs-field (which gives masses to the particles) and the Higgs-boson (which is the waste left of the Higgs-field after masses are given).
The Higgs-field can be considered a pair of complex scalar fields. These fields are supposed to have interactions with the other fields, something like c \bar \psi \Phi \psi with \psi being some fermion field and \Phi being (something like) the Higgs-field and c being the strength of the interaction. What the happens is that the state (of the world) we consider a vacuum is not the state in which \Phi = 0 but that e.g. \Phi = v. You can now define some new field which compensates this shift (and therefore is zero in the vacuum again): \phi := \Phi - v (note that \Phi is dynamic; v is a constant). If you substitute this new field into the original equation for the fields, the previous interaction term reads c \bar \psi ( \phi + v) \psi = \underbrace{c \bar \psi \phi \psi}_{I} + \underbrace{c v \bar \psi \psi}_{M}. The 2nd addend M is exactly what a mass term looks like in quantum field theory. In that sense, the Higgs-field is supposed to give mass to the particles (actually only for the fermions here, the terms for the gauge bosons are slightly different but it also results in that the parameter v gives them their masses). I is still an interaction term.
You now have expressed the original Higgs-field \Phi as the offset \phi from its value in vacuum. By a suitable redefinition of the gauge boson fields, three of the four degrees of freedom \phi has are absorbed into the newly-defined fields (let's call them "interaction particles"). There remains one degree of freedom that is not absorbed. This degree of freedom is the Higgs-boson.
In short: There is a thing called Higgs-field. This field is redefined and partially absorbed by other fields (via their redefinition). The remaining one real-valued degree of freedom of exitation from the vacuum-state is the Higgs-boson.
If Higgs particle is a boson then why this Higgs mechanism not considered a force?
In the end, every such question boils down to "it's a matter of definition and convention". The Higgs-field is different from the fields that (after the redefinition already mentioned above) later form the particles you probably know as force carriers. The "gauge fields" (that's their name) result from demanding that there exist some abstract symmetries ("gauge symmetries" - now guess where the name "gauge boson" comes from
). The Higgs-field is no such result but simply put into the Standard Model to make it work. Resulting from different approaches, the Higgs-field and the gauge fields are conceptually and practically quite different, the most obvious difference being the spin (scalar vs. vector), less obvious being that the unredefined Higgs-field must have mass terms (and even more) while the unredefined gauge fields are mathematically forbidden to have them.
In short: The Higgs-field from which the Higgs-boson stems is conceptually different from the gauge fields from which the interaction particles (photon,W,gluon,Z) originate.
