What is "concretely" $\int_a^b f(t)dN_t$ when $N_t$ is a Poisson process? What is "concretely" $\int_0^1 f(t)dN_t$ when $N_t$ is a Poisson process ? In the sense, what is the interpretation ? Is it something as $$\lim_{n\to \infty }\sum_{k=0}^{n-1}f(t_i)(N_{t_{i+1}}-N_{t_i}),$$
where $\{t_i\}$ is a partition of $[0,1]$ or as the Brownian motion, this sum doesn't converges ? And if yes, what could be the interpretation behind ?
 A: Lurking in the background of Associated with the Poisson process $N_t$ is the set of random times $T_k$ where $N_t$ has jumps.  The number of such times occurring in $[a,b]$ is random, with Poisson distribution $\lambda([a,b])$ where $\lambda$ is the intensity measure of $N_t$, and the distributions of the $T_k$ that do occur in  $[a,b]$ is iid with probability distribution proportional to $\lambda$ restricted to $[a,b]$.  In the special case where $\lambda$ is a scalar multiple of Lebesgue measure, your quantity $Q=\int_a^bf(t)dN_t$ is representable as $Q= f(T_1)+f(T_2)+\cdots+f(T_N)$ where $N$ is Poisson with expectation $\lambda|b-a|$ and where the $T_k$ are iid uniform on $[a,b]$.
Your $Q$ is thus a random quantity.  Its expectation is $\int_a^b f(t) d\lambda(t)$, or, in an alternate notation, $\int_a^bf(t)\lambda(dt)$.
You should, as an exercise, be able to work out a formula for the variance of $Q$ using the above representation.
A: The integral $\int_0^t f(t) dN(t)$ can be expressed as 
$$\sum_{k=0}^{N_t}f(T_k)$$ 
where $T_k$ is the time of the $k^{th}$ jump of the Poisson process. 
To understand the above expression note first that in interval $[0,t]$ the value of the Poisson process is equal to the number of its jumps; so there are $N_t$ jumps in this interval. Second, note that the integral is $0$ for all $t$ except for the times that the Poisson process jumps. At the time of the jump, the integral increases by $F(T_k)\Delta N_{T_k}=F(T_k)$. Note also that, as kimchi lover pointed out, this integral is a random variable; it will depend on the actual realization of the path of the Poisson process.
