If an object accelerates towards the speed of light many known rules of physics don't apply or at least have to be adapted to fit.
An easy "key" to the problem is that the speed of light is not relative. It is always constant regardless of the speed/location of the observer.
Let's say someone shoots a laser beam on the moon and watches the reflected signal from earth. He sees light, that travels to and from the moon. It travels twice the distance from earth to moon.
Someone in a really fast spaceship sees the moon and the light flying by. Thus, the light from earth to moon and the light from moon to earth are not on the same line but are tilted. A person in the spaceship sees some kind of V-shape for the way the light travels. Assuming that the speed of light is constant for everyone that leaves only one choice:
The light observed from the spaceship travels a longer distance. Therefore, the time has to go slower in the spaceship than on earth. 562
This phenomenon is known as time dilation. Similar things happen to distances.
The factor by which lengths/time are changed is called Lorentz Factor. It is calculated this way:
$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
This factor also applies to mass. The mass of not moving object is multiplied with this factor. If the value of v increases towards the speed of light c, the term under the square root will become smaller and eventually 0 for $ v = c $. The whole term $\gamma$ increases with increasing speed. The relative mass of a moving object with the mass $m_0$ is $m = \gamma \cdot m_0$.
Because of the increasing mass, more and more energy is needed to accelerate towards the speed of light and eventually you need an infinite amount to reach the speed of light. Therefore it is not possible to reach the speed of light through standard acceleration.
There's a theoretical approach for realizing speeds higher than the speed of light: The Alcubierre Drive. It basically deforms the space the traveller is in, so it doesn't move faster than the speed of light inside its space but seen from the outside it is faster. 343