Question: How can i find necessary and sufficient condition for this system to have a solution:

$x \equiv u\_1(mod \ m\_1)$

$...$

$x \equiv u\_n(mod \ m\_n)$

Answer: 
The Chinese remainder theorem states that if the $m\_i$ are co-prime a solution will exist. 

So it is sufficient that the $m\_i$ be co-prime.

If $x \equiv u\_i \mod m\_i$ and $x \equiv u\_j \mod m\_j$ and $\gcd(m\_i, m\_j) = d \ne 1$ then $x = k\_im\_i + u\_i = k\_jm\_j + u\_j$ so $u\_i - u\_m = d(k\_im\_i/d - k\_jm\_i/d)$ so $u\_i \equiv u\_j \mod d; x \equiv u\_i \mod \frac {m\_i}d; x\equiv u\_j \mod \frac {m\_j}d$ which will have a solutions as $\frac {m\_i}d$ and $\frac {m\_j}d$ are coprime.

So it is necessary and sufficient that if any $m\_i, m\_j$ are not coprime that $u\_j\equiv u\_i \mod \gcd(m\_i, m\_j)$.
Try looking at the proof of the Chinese Remainder Theorem to see how the case when the $m\_n$'s are coprime is dealt with and try to make a generalization.