Accelerating particles to speeds infinitesimally close to the speed of light?
By special relativity, the energy needed to accelerate a particle (with mass) grow super-quadratically when the speed is close to c, and is ∞ when it is c.
$$ E = \gamma mc^2 = \frac{mc^2}{\sqrt{1 - (\text{“percent of speed of light”})^2}} $$
Since you can't supply infinite energy to the particle, it is not possible to get to 100% c.
Edit: Suppose you have got an electron (m = 9.1 × 10-31 kg) to 99.99% of speed of light. This is equivalent to providing 36 MeV of kinetic energy. Now suppose you accelerate "a little more" by providing yet another 36 MeV of energy. You will find this this only boosts the electron to 99.9975% c. Say you accelerate "a lot more" by providing 36,000,000 MeV instead of 36 MeV. That will still make you reach 99.99999999999999% c instead of 100%. The energy increase explodes as you approach c, and your input will exhaust eventually no matter how large it is. The difference between 99.99% and 100% is infinite amount of energy.
There are (at least) two explanations, kinematical and dynamical.
Dynamics
When you want to make an object accelerate you have to use energy to produce force on the object. The force is $F = ma$ (this equation is not really correct in SR but it suffices for our purposes) Now the point of SR is that the mass $m$ that the object seems to be having when it is moving with respect to you is not constant. It goes like $m = m_0 \gamma(v)$ where $m_0$ is the objects invariant mass (as seen from its own rest frame) and $\gamma(v)$ is the Lorentz factor. Now $\gamma(v) \to \infty$ as $v \to c$. So this means that the (apparent or relativistic) mass of the object becomes arbitrarily large and you would need an infinite amount of energy to get to the speed of light.
Kinematics
From the kinematical point of view it all boils down to relativistic concept of velocity. In SR when you want to change particle's speed you have to boost it. This is described by a certain Lorentz transformation.
Now its useful to move to the dual point of view. Instead of saying that you boost the particle you can just change your reference frame in the opposite way. So instead of giving the particle speed $v$ in direction $\mathbf x$ you will look at the particle at rest from a reference frame that has speed $v$ in direction $-\mathbf x$. This transformation is also described by a Lorentz transformation.
Now every Lorentz transformation preserves the relations $v < c$, $v = c$ and $v > c$ (the middle one is actually Einstein's postulate on invariance of speed of light in every inertial frame). That means that if your velocity is less than speed of light it will be so in any reference frame. And also that if some particle was once going slower than speed of light, it will always do so.
You have to understand special relativity. It's basically because Newtonian mechanics breaks down at speeds close to the speed of light and $F=ma$ is false. It's basically because your mass isn't constant, it varies based on your speed. And as you approach $c$, your mass has to approach infinity and thus you'll need infinite force to move accelerate from $c - {\Delta}v\to c$.
This is just a basic overview, I'm sure someone will come with a much more detailed overview, but you can take a look at the Wikipedia entry on SR, specifically the part about relativistic mechanics.