Intuitive meaning of Exact Sequence I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all.
Can anyone explain them for me? Thanks in advance.

It's sometimes a good idea to think of long exact sequences in the more general context of chain complexes. A chain complex $C$ of maps $d_i$ is a sequence $$\cdots\rightarrow A_{i+1}\stackrel{d_{i+1}}{\rightarrow}A_i\stackrel{d_i}{\rightarrow}A_{i-1}\rightarrow\cdots$$ such that $d_i\circ d_{i+1}=0$ for all $i$. Now, from this very mild condition, we can do quite a lot with this chain complex. In particular, we know that $\mbox{im}\, d_{i+1}\subset\ker d_i$ and so we can take a quotient.
Let $$H_n(C)=\ker d_n/\mbox{im}\,d_{n+1}.$$ We call this the $n$th homology of the chain complex $C$. The homology of a chain complex is an extremely powerful tool and motivates much of homological algebra.
We might like to ask then, what conditions on the chain complex $C$ would tell us that its homology in every degree is trivial? It turns out that the homology of $C$ is trivial in every degree if an only if $C$ is an exact sequence. This isn't hard to show as, if $C$ is an exact sequence then $\mbox{im}\, d_{n+1}=\ker d_n$ and so $H_n(C)=\ker d_n/\mbox{im}\,d_{n+1}=0$. Also, if $H_n(C)=0$ then $\ker d_n/\mbox{im}\,d_{n+1}=0$ and so $\mbox{im}\, d_{n+1}=\ker d_n$.
