I must premise that I am not a specialist in numerical analysis, therefore I may be not right when talking about more popular methods in this field pertaining the solution of ITEs. Said that, however, I think I can be of some help.
Could you recommend me any articles or book with a brief overview of some methods (maybe classical one), please?
Perhaps a good reference is the book by Prössdorf and Silbermann : they use (and advocate) a Boundary Element Method Based approach which uses polynomial and spline approximation methods for approximating the sought for solution. The emphasis of the text is mainly on integral equations, but also integrodifferential operators in the form of pseudo differential operators are the topic of the whole chapter six.
Which methods are most popular?
Notwistanding what I said above, it seems to me that methods based on polynomial approximation have some popularity, especially for $(n>1)$-dimensional ITE: this is due to the fact that there are cubature formulas for the approximate calculation of integrals which are exact on given classes of polynomials enjoying also nice approximation properties. This allows a mitigation of the effects of the so called "curse of dimensionality".
 Siegfried Prössdorf and Bernd Silbermann (1991), Numerical analysis for integral and related operator equations. Licensed ed. (English), Operator Theory: Advances and Applications. 52. Basel-Boston-Berlin: Birkhäuser Verlag, pp. 542, ISBN: 3-7643-2620-4, MR1193030, Zbl 0763.65103.