I'm somewhat confused over the definition of Artinian and Noetherian rings.
A Noetherian ring is a ring in which there are no infinite chains of nested ideals. That is, if $I_i$ are some ideals in a ring $R$, which satisfy the condition $I_1\subset I_2 \subset ... \subset I_n \subset...$, then $\exists$ integer $N$ such that $I_N = I_m$ for all $m\ge N$.
An Artinian ring is a ring in which there are no infinite chains of nested ideals, but... if $I_i$ are some ideals in a ring $R$, which satisfy the condition $I_1\supset I_2 \supset ... \supset I_n \supset...$, then $\exists$ integer $N$ such that $I_N = I_m$ for all $m\ge N$.
But what is the difference between the Noetherian and the Artinian conditions? Can't we rewrite the ascending chain in the Noetherian case as a descending chain, and vice versa?
No, you can't just rewrite an infinite ascending chain as an infinite descending chain or vice versa. For instance, suppose you have an ascending chain $$I_1\subset I_2 \subset \dots \subset I_n \subset\dots$$ and want to get a descending chain $$J_1\supset J_2 \supset \dots \supset J_n \supset\dots$$ What are you going to define $J_1$ to be? You can't make it $I_1$, since that's the smallest ideal in the chain, and $J_1$ needs to be the largest instead. If you made $J_1$ be $I_2$, you could then make $J_2$ be $I_1$, but then you wouldn't be able to define $J_3$. If you made $J_1$ be $I_3$ you could define $J_2=I_2$ and $J_3=I_1$, but then you would have no way to define $J_4$. And so on.
For a similar phenomenon that should be familiar, there is an infinite ascending chain of natural numbers, namely $0\leq 1\leq 2\leq 3\leq\dots$. But there is no infinite descending chain of natural numbers, since if you start such a chain at $n$, you can only ascend $n$ times before you run out of natural numbers!