Marxism
Marxism, simply put, is the political belief that the human journey toward progress happens with the occasional economic and political revolution of the lowest class against the interloping "Bourgeoisie." Although he eventually moved on from this view, this view defined his young life. Marxism was Kropotkin's response to his nobles oblige, or the responsibility he felt as a uniquely privileged person, having been raised in luxury and private schools, to try and use his education for good, specifically for the good of the lowest class, the very poor.
Anarchism
When Kropotkin became enlightened to his fully formed, mature political views, he made a switch from the academic realm of Marxism and the progress of the human race through government, to the radical opposite of that view—absolute anarchy. His anarchist philosophy landed him a place in history as one of the most important voices of the anarchist movement. The philosophy is not nearly as political as Marxism was, but it was way more offensive, especially to the governments who arrested him for sharing his views with the public.
Political activism
No doubt, Russian politics circa 1900 are fascinating, but that doesn't mean that today's modern politic is any less important. The central idea of this entire book is that this man was deeply passionate about his civic duty. He felt that his job as a human being was to stand for the principles that mattered most to him, and although it's easy to point at the shortcomings of his political views, no one can say he was not a right-fighter. Several times he was imprisoned for speaking his political view, so if nothing else, he is a martyr for freedom of speech.
