Simple grid drawing in TikZ

documentclass{standalone}
usepackage{tikz}
begin{document}
    begin{tikzpicture}
        tikzset{sphere/.style={
            draw,
            thick,
            #1!75!black,
            ball color=#1,
            circle,
        }}
        tikzset{arrow/.style={
            very thick,
            ->,
            >=latex
        }}
        draw[fill=red, draw=none] (2, 1) rectangle (3, 2);
        draw[fill=red, draw=none] (1, 2) rectangle (2, 3);
        draw[thick] (0, 0) grid (4, 4);
        node[sphere=blue, minimum size=0.5cm] (projection) at (1.5, 2.5) {};
        node[sphere=blue, minimum size=1cm] (ball) at (0.8, 5) {};
        draw[arrow, blue] (ball.south) -- (projection.north west) node[pos=0.15, right, color=black] {footnotesize Projection};
        node at (1.15, 2.15) {scriptsize(p_j)};
        node[below left] at (ball.south west) {(n_j)};
        
        begin{scope}[xshift=6cm]
            draw[fill=red, draw=none] (2, 1) rectangle (3, 2);
            draw[fill=red, draw=none] (1, 2) rectangle (2, 3);
            draw[thick] (0, 0) grid (4, 4);
            node[sphere=blue, minimum size=0.5cm] (projection) at (1.5, 2.5) {};
            node[sphere=blue, minimum size=1cm] (ball) at (0.8, 5) {};
            draw[arrow, blue] (ball.south) -- (projection.north west) node[pos=0.15, right, color=black] {footnotesize Projection};
            node at (1.15, 2.15) {scriptsize(p_j)};
            node[below left] at (ball.south west) {(n_j)};
            node[sphere=green, minimum size=0.5cm] (projection2) at (2.5, 1.5) {};
            node[sphere=green, minimum size=1cm] (ball2) at (3.2, 5) {};
            node at (2.825, 1.15) {scriptsize(p_{j'})};
            node[below left] at (ball2.south west) {(n_j)};
            draw[arrow, green] (ball2.south) -- (projection2.north);
            draw[arrow] (projection2.north west) -- (projection.south east);
        end{scope}
    end{tikzpicture}
end{document}

The key commands used here are

draw (a, b) grid (c, d);

draws a grid from (a, b) to (c, d).

fill[<colour>] (a, b) rectangle (c, d);

draws a rectangle with diagonally opposite corners (a, b) and (c, d) rectangle has no boundary and is filled with the colour <colour>.

node[circle, ball color=<colour>, minimum size=xcm] at (a, b) {};

draws a circle of colour <colour> and diameter x measured in centimeters at (a, b).

I also made use of nodes and coordinates, in particular if you do

node (name) at (a, b) {};
draw (name.south) -- (c, d);

then this places a node at (a, b) and calls the node name. The second command then draws a straight line from the bottom of node (.south other compass directions work similarly) to the point (c, d).