There isn't a simple answer to your problem because it depends on what you mean by mass.
You've probably heard of Einstein's equation $E = mc^2$, and if you rearrange this equation you can use it as a definition of the mass:
$$ m = \frac{E}{c^2} $$
For a stationary object the mass we get is the rest mass or invariant mass, which is what we normally mean by mass. However if the object is moving then it has some kinetic energy so:
$$ E_{total} = mc^2 + \text{kinetic energy} $$
and we can define a relativistic mass using:
$$ m_r = \frac{E_{total}}{c^2} $$
and for a moving particle this relativistic mass is obviously greater than the invariant mass. This is mathematically a possible definition for the mass, but in the 109 years since special relativity was formulated it has caused generations of students much confusion, just as it is confusing you now in 2014. To avoid this we now rarely refer to relativistic mass, and modify Einstein's equation to be:
$$ E^2 = p^2c^2 + m^2c^4 $$
where $m$ is the invariant mass and $p$ is the momentum. Though I won't duplicate it here, the Wikipedia article on Mass in special relativity gives a proof that the two equations are the same.
You ask if the mass increase is real, but that can't be answered because you have to define what you mean by real. The difference between relativistic and invariant mass comes from accounting for the energy in different ways. One is no more real than the other, though the concept of invariant mass is much less confusing.