Binary stars system ultimate fate I read an article mentioning that when two stars orbiting each other they'll produce gravitational waves that can carry away energy into space. These phenomenon increases the speed of the two orbiting stars and draws them closer to each other until they finally collided, is it true that gravitational waves can carry away energy into space?
Well, yes apparently that is how it works and a Nobel prize was won in 1993 by Hulse and Taylor for demonstrating this phenomena in the object PSR B1913+16.

PSR B1916+16 is a pulsar (a rapidly rotating neutron star with a strong magnetic field that emits beamed radiation along its magnetic axis). The pulsar provides an extremely precise ticking clock with which to study the motion of stars in a binary system. The pulsar is in orbit with a companion that is also a neutron star (though not an observable pulsar). The orbit is elliptical with a period of 7.75 hours.

The period of the orbits is decreasing. That is, the two stars are gradually getting closer together. This effect is a consequence of general relativity, that predicts that such massive objects that are so close to each other will act as a source of gravitational waves. The gravitational waves have not (yet) been directly observed, but GR makes a prediction for how much energy they carry away and hence at what rate the orbital period changes.

The rate of decrease in the distance between two bodies is given by
$$\frac{dr}{dt} = -\frac{64G^3}{5c^5}\frac{(m_1 m_2)(m_1+m_2)}{r^3}$$ 
and leads to a merger between the two objects in a time of
$$t = \frac{5c^5}{256G^3}\frac{r^4}{(m_1 m_2)(m_1+m_2)}, $$
where $m_1$ and $m_2$ are the masses of the two orbiting bodies, $G$ is the gravitational constant and $c$ is the speed of light.

In slightly more friendly units
$$ \frac{dr}{dt} = 7.8\times10^{-19} \frac{(M_1 M_2)(M_1+M_2)}{r^3}\ au/yr,$$
where $r$ is in astronomical units and the masses are in solar masses, and
$$ t = 3.2\times10^{17} \frac{R^4}{(M_1 M_2)(M_1+M_2)}\ yr. $$

Thus you can see that the effect gets much stronger (and the merger times shorter) if you have two high mass objects (it depends on the product of the masses) with a small orbital separations.

For "normal" stars you can never get them closer than their stellar radii and generally means that the effect is too small to be important. However, compact objects (white dwarfs and neutron stars) can be brought much closer together and this is why the effect has been seen there. The Hulse-Taylor binary components 
have masses of $\sim 1.4 M_{\odot}$ and are separated by an average of about 2 million km (0.013 au). Inserting this into the formula above give a merger time of 1.7 billion years. A more accurate calculation that takes account of the elliptical orbit gives 300 million years.

However, if you made the neutron stars orbit with a separation of 0.1 au (more typical for a normal stellar binary system), the merger timescale would increases to well beyond the current age of the universe.