What is the maximal path of a tree? Could anyone explain obviously what the maximal path is ? Is it necessary for a tree that has two maximal paths that share no common vertex ?
I will assume that you meant a connected tree. Also, I am using maximal to mean "of maximal length," which may not be what you meant.

A path in a tree is a sequence of vertices, with no repeats, where successive vertices in the sequence are adjacent. If a path $P$ is at least as long as any other path, then that path is maximal. It is possible for there to be more than one maximal path. For example, in this tree,

     C
    /
A--B
    \
     D


both of the paths $A,B,C$ and $A,B,D$ are maximal, since they are of length $3$, and there are no paths of length $4$ or greater.

Note that these two paths share a common vertex. This is always true: any two maximal paths will share a common vertex. 

Why? Suppose you had two disjoint paths, $P_1=v_1,v_2\dots,v_n$ and $P_2=w_1,w_2\dots, w_n$. Consider the pair $v_i,w_j$ for which the distance between $v_i$ and $w_j$ is minimal (here, distance means the length of the shortest path connecting them). Then there is a path $v_i,p_1,p_2,\dots,p_k,w_j$ which connects $v_i$ and $w_j$. None of the intermediate vertices $p_1,\dots,p_k$ can be in $P_1$ or the $P_2$, because of the minimality when we chose $v_i$ and $w_j$. This implies that $P_1$ and $P_2$ weren't maximal, since at least one of the following paths are longer:
$$
v_1,\dots,v_i,p_1,\dots,p_kw_j,\dots,w_n
$$
$$
w_1,\dots,w_j,p_k,\dots,p_1,v_i,\dots,v_n
$$
We have shown that disjoint paths are not maximal, which means that any two maximal paths intersect.
Maximal path in a graph G is a path P in G that is not contained in a longer path.

When a graph is finite, no path can extend forever, so maximal(non-extendible) paths exist.