Inflation is a name for a family of theories that which are essentially modifications of the original Big Bang theory. Inflation states that our universe began as a hot dense ball of matter and energy, then exploded much more rapidly than the speed of light, doubling in size every 10-34 seconds.
A curiosity of quantum physics is that it allows our universe to be created out of nothing. Particles are created and rapidly disappear all the time in a process called "vacuum fluctuation." The less energy a particle has, the longer it can exist. Luckily for us, our universe has a net energy of zero. This is because all the positive energy in our universe is cancelled out by gravity. It works like this: our sun is a ball of extremely hot gas. This gas is so dense and hot that it is set off in a set of fusion reactions, producing even more heat. Our sun is trying to expand because of this heat, but gravity is working to collapse the sun. Because everything balances out, our sun has neither expanded until it stops burning, nor collapsed in on itself. Our universe is the same way, with all the energy locked up in matter cancelled out by gravity.
The universe needed inflation to get started because when it first appeared, it was a tiny ball of matter and energy 10-35 meters across, about 1020 times smaller than a proton. Gravity would have instantly crushed such a dense orb if something had not given the universe a push...inflation.

Inflation caused the universe to expand too fast for gravity to snuff it out of existence. It works like this: in the old model of the Big Bang, the universe could only expand at the speed of light, about 670 million miles an hour. This means that after two ticks of some imaginary clock, the universe had doubled in size. After four ticks, the universe had doubled in size again, and again after 8 clicks. But the inflationary model states that the universe doubled in size after every tick of the clock. After 4 ticks, the universe had doubled in size 4 times, after 8 ticks, it had doubled 8 times. This means that the universe expanded much faster than the speed of light, doubling every 10-34 seconds, or every 0.00000000000000000000000000000001 seconds. After about 10-32 seconds, the universe slowed down to the speed of light, but by then the universe had expanded from a size smaller than a proton to about the size of a grapefruit.
After this short period of exponential expansion, the universe can be described by more traditional physics as a swirling ball of hot gas and energy expanding at the speed of light.