Every person in the world of Fairy Tail has a container inside them that contain the Magical energy of the earth. Those who can tap into this energy source and use it are Mages. When a Mage uses this energy they produce spells and the container slowly empties until they run out and have no more Magic power. Over time the container refills itself and the Mage can once again cast spells.
Now, Second Origin is like a second container in the body. Most Mages can only tap into their first container, and when it runs out they are left Magic-less until it refills. However, powerful Mages who use a lot of Magic power to cast spells can tap into this power source, and hence become incredibly strong.
In chapter 263, Ultear revealed that due to her time Magic she can alter whether or not this second container is open. She used her Magic on those in Fairy Tail so that instead of having to train to open this container, they merely had their time advanced so they could tap into it instantly. Apparently though this method of obtaining Second Origin is very painful :P
As such, the Mages can now cast spells of a much higher power and ability than they could before, as we are currently seeing in the Grand Magic Games with Lucy's multiple summon and Gray's new techniques.
And that's that, I hope this made sense XD