These are two examples from the book. Why in the first one it's neglecting \$i_{2}\$ ,shouldn't the diode's current be found with Kirchhoff's first law like in the second example? How can I know when to consider the second current and when not? I know that the diode's current get's very large for forward voltage but why the current isn't neglected in every parallel circuit like in these two.
Why in the first one it's neglecting \$i_2\$?
Because if you take \$V_i\$ as an independent variable, \$i_2\$ doesn't change \$V_o\$. \$V_o\$ depends only on \$V_i\$, R, and R1.
How can I know when to consider the second current and when not?
You can calculate what the second current is (\$V_i / R_2\$ in this case) and then see whether you ever use that information in calculating whatever final result you're going for (\$V_o\$ in this case).
why the current isn't neglected in every parallel circuit like in these two.
Because sometimes it matters. If the source providing \$V_i\$ had an internal resistance, for example, then \$i_2\$ would contribute to the drop across that internal resistance, so you'd have to know it to find the output voltage.
