The following paragraph is taken from a book on Lie groups, but I am not understanding the description.
A linear transformation $X$ of a vector space $E$ over real or complex can be though of as a vector field: $X$ associates to each $p$ in $E$ the vector $X(p)$.
For example, consider fluid motion.
The velocity of the fluid particles passing through the point $p$ is always $X(p)$. [it is always $X(p)$, or we are denoting it by a vector $X(p)$?]
The vector field is then the current of the flow [what it means?], and the paths of the fluid particles are trajectories.
Consider the trajectory of the fluid particle that passes through a certain point $p_0$ at time $t=t_0$.
If $p(t)$ is its position at time $t$, then its velocity at this time is $p'(t)=d p(t)/dt$.
Since the current at $p(t)$ is supposed to be $X(p(t))$ [why?], one finds that $p(t)$ satisfies the differential equation (how?) $$ p'(t)=X(p(t)) $$ together with the initial condition $p(\mbox{at }t=0)=t_0$.
I searched in Google the term "current in fluid", but I found only Wiki link, with only definition, but no explanation/examples. So I was unable to proceed in the understanding of above paragraph.
Also, I am away from Mathematical Physics from more than 15 years, so it would be great help if one can explain it.
The book's description is not great, to be honest. You will occasionally hear the term "fluid current" used, but it's not that common. Really, the vector field $X$ represents the velocity of the fluid. So $X(p)$ is the velocity of the fluid at a point $p$ (i.e. the velocity of a particle of the fluid at the point $p$). Note in this case, since $X$ does not explicitly depend on time $t$, one is assuming that the velocity field $X$ is time independent, i.e. the flow is steady. You can also have the case that the velocity field depends on time, in which case the velocity of the fluid at point $p$ and time $t$ would be $X(p,t)$.
Now suppose you track the position of one specific fluid particle over time, and denote that by $p(t)$. Its velocity at time $t$ is just $p'(t)$. But by assumption, the velocity of the fluid at $p(t)$ is $X(p(t))$. These must be equal, which is where the equation $p'(t) = X(p(t))$ comes from. Mathematically, the curve $t\mapsto p(t)$ is called an integral curve of the vector field $X$.