How do I understand the main concepts in a paper by just skimming through it? What's the best method to understand the real concept that a paper is trying to explain?

Should one go through the paper as it is presented, from first page to last page? Is there a specific walk-through to understand it faster, such as first reading the result section, followed by the concept section, and then discussion/conclusion section? Is there a best practice to be followed?
If the paper is well written, then the abstract should tell you most of the story at a high level. Then the introduction should give more technical coverage again of the whole story – you should then know the problem and the results obtained, though not all the details. The remainder of the paper then will contain just the details.

Of course, most papers do not do that, so you may also need to read the conclusion, the discussion section, and perhaps any other introductory sections. In the end, you may need to read the whole paper before you get what it is about. 

However, to really understand the paper, you will need to go through it very slowly, with pen and paper at hand, and try to replicate the reasoning/ideas given in the paper, filling in the gaps – possibly with the help of additional literature.

Not all papers need to be read so deeply. Reading just the abstract and maybe the introduction and conclusion should help you decide how deep to read the paper.