C#: LINQ and IEqualityComparer

When writing programs in C# with the help of LINQ I have come across the IEqualityComparer<T> generic interface several times. The name conveys the purpose of the interface clearly, still, surprisingly, my first attempts to apply it were marked by strong confusion - the thing merely didn't work the way I wanted it to. Moreover, after going to Google and StackOverflow I have noticed that I'm not the only one to face difficulties with it. After inspecting multiple SO questions and answers related to the topic as well as some other articles I have both found the solution that fitted my needs for the moment and got some understanding of the way IEqualityComparer is actually used by LINQ operators. Here I will try to explain what I got from there.