Whether it's overcoming the incessant urge to hit snooze again, or pushing to finally complete that manuscript, you and I cross barriers everyday. The very same is true of the most fundamental natural phenomena.
For example, consider the all too familiar rubber band. Pretend you're slowly stretching a rubber band, the potential energy you're giving the rubber band is straining the internal connections that maintain its structure. Eventually, if you stretch enough, you'll overcome these connections and the rubber band will break - converting the potential energy you've applied into kinetic, in the form of a 'snap.' Figure 1, hopefully, gives you a crude visualization of this.
Figure 1: Potential energy of a rubber during stretching.
This process may be viewed as a form of barrier crossing. Here, the rubber band is continuously strained until it cannot take anymore and finally lapses, which is coincident with crossing its 'barrier.'
Now lets consider how this analogy extends to the molecular world. Lets pretend we're tracking the movement of some particle. This particle is in some sort of box being constantly bombarded by neighboring particles, also, in the middle of the box there is 'hill.' Our particle starts on one end of the box, and if it's hit in just the right way, it may finally overcome the hill and end up on the other end of the box. Figure 2 shows a brief animation of this process with a gaussian hill, the faded pink line shows the value of the derivative at each point.
Figure 2: Animation of a particle crossing a gaussian barrier. Pink line represents the value of the gaussian's derivative.
In this animation I'm influencing the particles movement by biasing it's motion to the right (similar to how you may bias a rubber band by stretching it). What's hindering the particles movement is the slope of the 'hill.' The particle struggles most to overcome the point of maximum slope (highest value of the pink curve), but once traveling past the gaussian maximum it cruises to the end - like ball might after crossing the peak of a hill.
This barrier crossing is directly linked to the idea of 'Quantum Tunneling.' Now, if you're unfamiliar with quantum mechanics this epigram may seem bit mysterious to you. What quantum tunneling refers to is crossing the barrier without actually moving across the barrier. Imagine having a ball halfway up a hill, and all of a sudden, the ball magically appears on the other side of the hill - that's quantum tunneling. This phenomena might seem to be physics magic, but is simply a result of the random sampling. If you're working with microscopic particles on the time scale of femtoseconds, a random 'kick' that moves particle across the barrier is a very serious concern.
You may be asking yourself at this point, "Kirill, this is ridiculously, I've never seen any object move through a hill or pass through a wall." In fact, this concern is the principal limitation in making smaller transistors. Although not evident in our macroscopic world, these quantum limitations are an ever important limitation that effect our every day lives.