Expanding the comment of Donu Arapura, let $X$ be a variety and $Y\subset X$ a subvariety. Then, you have a short exact sequence of sheaves $$ 0\to\mathcal I_Y\to\mathcal O_X\to\mathcal O_X/\mathcal I_Y\to 0, $$ where $\mathcal I_Y$ is the ideal sheaf of $Y$. By definition, $\mathcal O_X/\mathcal I_Y=\mathcal O_Y$ is the structure sheaf of $Y$.

If $\mathcal F$ is any invertible sheaf, then tensoring by $\mathcal F$ leaves the sequence exact, so that you have a short exact sequence $$ 0\to\mathcal I_Y\otimes\mathcal F\to\mathcal F\to\mathcal O_Y\otimes\mathcal F\to 0 $$ and $\mathcal O_Y\otimes\mathcal F$ is juste the restriction of $\mathcal F$ to $Y$.

Now, suppose that your $Y=D$ is a (Cartier) divisor, and $\mathcal F=\mathcal O_X(D)$ is its associated (invertible) sheaf of sections (meromorphic functions with poles allowed along $D$). In this case, $\mathcal I_D=\mathcal O_X(-D)$ and the above-mentioned short exact sequence becomes $$ 0\to\mathcal O_X\to\mathcal O_X(D)\to\mathcal O_D(D)\to 0, $$ and $\mathcal O_D(D)$ is nothing but the restriction $\mathcal O_X(D)\otimes\mathcal O_D$ of the invertible sheaf $\mathcal O_X(D)$ to the hypersurface $D$.

You can argue dually for $\mathcal O_D(-D)$, which is thus just the restriction to $D$ of the invertible sheaf $\mathcal O_X(-D)$.

So your "second map", is just the restriction map.

For your last question, an heuristic explanation is the following: blow-up a smooth point on a surface to obtain a new surface $\widetilde X$, and call the exceptional divisor $E$. Then $\mathcal O_{\widetilde X}(-E)$ restricted to $E$, which is precisely $\mathcal O_E(-E)$, can be easily shown to be isomorphic to the (anti)tautological line bundle $\mathcal O(1)$ over $\mathbb P^1\simeq E$ (you can find that on every introductory book in algebraic geometry). Now, you are blowing up a singular point which is an isolated quotient singularity of order three, thus in some sense you are "counting three times" your point, so that $\mathcal O_E(-E)$ now becomes isomorphic to $\mathcal O(3)$.

Expanding the comment of Donu Arapura, let $X$ be a variety and $Y\subset X$ a subvariety. Then, you have a short exact sequence of sheaves $$ 0\to\mathcal I_Y\to\mathcal O_X\to\mathcal O_X/\mathcal I_Y\to 0, $$ where $\mathcal I_Y$ is the ideal sheaf of $Y$. By definition, $\mathcal O_X/\mathcal I_Y=\mathcal O_Y$ is the structure sheaf of $Y$.

If $\mathcal F$ is any invertible sheaf, then tensoring by $\mathcal F$ leaves the sequence exact, so that you have a short exact sequence $$ 0\to\mathcal I_Y\otimes\mathcal F\to\mathcal F\to\mathcal O_Y\otimes\mathcal F\to 0 $$ and $\mathcal O_Y\otimes\mathcal F$ is just the restriction of $\mathcal F$ to $Y$.

Now, suppose that your $Y=D$ is a (Cartier) divisor, and $\mathcal F=\mathcal O_X(D)$ is its associated (invertible) sheaf of sections (meromorphic functions with poles allowed along $D$). In this case, $\mathcal I_D=\mathcal O_X(-D)$ and the above-mentioned short exact sequence becomes $$ 0\to\mathcal O_X\to\mathcal O_X(D)\to\mathcal O_D(D)\to 0, $$ and $\mathcal O_D(D)$ is nothing but the restriction $\mathcal O_X(D)\otimes\mathcal O_D$ of the invertible sheaf $\mathcal O_X(D)$ to the hypersurface $D$.

You can argue dually for $\mathcal O_D(-D)$, which is thus just the restriction to $D$ of the invertible sheaf $\mathcal O_X(-D)$.

So your "second map", is just the restriction map.

For your last question, an heuristic explanation is the following: blow-up a smooth point on a surface to obtain a new surface $\widetilde X$, and call the exceptional divisor $E$. Then $\mathcal O_{\widetilde X}(-E)$ restricted to $E$, which is precisely $\mathcal O_E(-E)$, can be easily shown to be isomorphic to the (anti)tautological line bundle $\mathcal O(1)$ over $\mathbb P^1\simeq E$ (you can find that on every introductory book in algebraic geometry). Now, you are blowing up a singular point which is an isolated quotient singularity of order three, thus in some sense you are "counting three times" your point, so that $\mathcal O_E(-E)$ now becomes isomorphic to $\mathcal O(3)$.