Trouble understanding Joint Probability Density for Discrete Random Variable

Question

I'm having trouble with understanding this topic. Here's 3 extracts from my book:

Does it mean that $P(X =x, Y = y) = p_{X}(x) \cdot P_{Y} (y)?$ Because, isn't this when the 2 R.V are independent? Also, from Definition 3.7.2 above, is $p_X(x) $ and it's $y$ counterpart a marginal pdf? If you could explain it in simpler terms, it would mean a lot since the notations throw me off.

Thanks