Because the stars are so far apart, proposed means of interstellar travel either involve very long trips, or very fast means of getting there. We'll focus on the fast means, because nobody likes to wait a long time to get to their travel destination.
There are four possible means of interstellar travel that are taken seriously. They are nuclear pulse propulsion — one of the first proposed, fusion-powered rockets, light sails propelled by lasers, and antimatter-based propulsion. Of these, antimatter propulsion and light sails seem the most promising as they are the fastest. Antimatter or light sails could reach speeds similar to that of light, taking advantage of the relativistic effect of time dilation, which could make the interstellar trip to Alpha Centauri, which is four light years away, seem like less than a year in duration.
The remaining methods of interstellar travel, nuclear pulse propulsion and fusion rockets, are limited to speeds of about 10% of that of light at max, making them of less interest. However, nuclear pulse propulsion is interesting because it has been within our grasp since 1945. If an aggressive nuclear space program were pursued in the 1950s and 1960s, humanity probably could have sent manned expeditions to the outer solar system by the 1970s. However, nuclear explosions are forbidden by current international law, and the Outer Space Treaty forbids the deployment of nuclear weapons in space.
One of the biggest requirements for interstellar travel is being able to slow yourself down once you reach your target star system. At speeds close to that of light, the braking process has to begin about halfway through your trip. One of the most discussed possible methods of interstellar travel - light sails accelerated by huge space-based lasers - would need to pack some nuclear bombs, a fusion reactor, or antimatter, to ensure that they have a means to slow down upon reaching their destination. For probes, fly-bys may be sufficient, but for serious colonization efforts, the ability to brake is a must.
One of the dead but famous ideas for interstellar spaceships is the Bussard ramjet, which would suck in and then burn interstellar hydrogen with a gigantic scoop, allowing it to get around carrying its own fuel. However, subsequent calculations found that the scoop would not be practical - the drag on it would exceed its thrust.