Arguably the technologies that most change the lot of mankind are the ones that take people or goods from one place and move them to another place. In the beginning, we got around by walking. Walking is fine for short distances, but it took humanity about 15,000 years to reach Tierra del Fuego from east Africa using walking as the mode of transportation. That's not very efficient.
Later people learned how to domesticate horses and other beasts of burden. Horse greatly opened up the distance a person could cover in a day, which expanded opportunities for hunting and trade. Using animals to carry heavy things is also easier than doing it yourself.
Later boats were invented to cross water. The first were probably just rafts to cross rivers and small lakes. In addition to fishing, boats also open up territory, since you are not slowed by terrain as you would be on land. Later, larger boats patrolled coastal waters, and eventually ships were made with sails that could cross oceans. Ships became faster, outfitted with steam and then diesel or nuclear engines. Faster ships expanded distance you could travel comfortably, which made the world even smaller.
Eventually airplanes were invented, and now we can cross between continents in a matter of hours. By using technology to greatly increase the distance we can travel, we have opened up new opportunities.
We are now at a bit of an impasse. True, we hang about in low earth orbit and have been to the moon, but that's about it. By mixing chemicals together and igniting them, we can travel from the earth to the moon in about 3 days. However, if we want to go anywhere else, we need better engines. A trip to Mars or the asteroid belt could take a year or more, and Jupiter and Saturn are many years away using chemical rockets. With travel times that long, it's hard to imagine actually doing it. Therefore, better engines will be required. Enter the electrical rocket!
A plasma thruster being tested. Source: Wikipedia
Using electricity to power rockets isn't really a new idea, but so far it hasn't been used for human travel. Some satellites use it for station keeping, and some probes use it for propulsion. It is weak, but can burn for a long time, adding up to huge accelerations. Here's how it works:
First, a gas is used. A powerful radio generator then attacks the gas, ionizing it into plasma. Powerful electromagnets then concentrate and heat up the plasma, and expel it from the ship. This produces thrust, which moves the ship forward.
Now, a chemical rocket will produce much more thrust at any given time. However, a chemical rocket is not nearly as efficient, so it burns through its fuel much more quickly. If one wanted to use chemical fuels to go to Mars, you would point your ship in the appropriate direction, do a short burn of your thrusters, and then coast all the way there. This requires a lot of fuel to be consumed.
With the plasma engine, you would point your ship in the appropriate direction and burn your engines. However, they would stay lit. Due to the enormous efficiency of the electric engine, you could afford to leave it on. If you were racing a chemical rocket, it would quickly leave you behind, until it stopped its burn. It would then simply coast.
With the electric engine, the burn wouldn't stop. You'd keep on accelerating, going faster and faster with each passing moment. Eventually, you would catch up to the chemical rocket and soon leave it far behind.
Over short distances, say from the earth to the moon, a chemical engine will always win. For very long trips, say to Jupiter, efficiency and speed are key. You can only bring so much fuel, and you need to go very fast in order to get there in a reasonable time. The slow but steady plasma thruster is the only thing that could accomplish both of those goals.