A worldline is a line traced out in spacetime by a point particle that holds only one position at any given instant. A worldsheet is the analogous two-dimensional surface traced out by a string traveling through spacetime. The worldsheet of an open string (with loose ends) is a strip; that of a closed string (a loop) is a cylinder. A slice through any of these reveals the particle or string's position at any given time. Generally, the term "worldline" is used to apply to the trace through time of particles, strings, or to other objects. The idea of worldlines was originally pioneered by Einstein and is now widely used to demonstrate concepts involving extended periods of time.
Worldlines are, essentially, graphs of time versus distance. For example, the image at left depicts the worldline of someone who slept till eight AM, arrived at work at nine AM, sat at a desk until lunch at noon, went for lunch, returned a half-hour later, left at three PM, got home at four PM, and immediately went back to bed. If this sequence of events sounds long, the graph rearranges it simply and comprehensibly. For much longer sequences of events, worldlines are powerful modeling tools.
However, worldlines should not be considered simply time/distance graphs, which of course are manmade for the purpose or information organization. While representations of worldlines certainly serve this purpose, worldlines themselves are real physical components that prove quite important to some aspects of string theory.
Worldlines have several important properties. First, a worldline can in theory loop back on itself, implying that time travel is at least theoretically possible. However, since what has already happened is encoded in the worldline, events in the past cannot be changed. Second, worldlines cannot be cut. The movie Back to the Future is a good example: a young man accidentally diverts his mother's attention from his father, seemingly ensuring his own disappearance since his parents will not marry and he will never be born. However, this implies changing an event in the past and causing the young man's disappearance. Since worldlines cannot stop abruptly, the scenario is impossible. Third, worldlines cannot materialize spontaneously; they trace out the path of matter through spacetime, and matter cannot be spontaneously generated, so it follows that worldlines cannot simply appear from nothing at all. Fourth, worldlines can combine. This happens every time a baby is born - the worldlines of the father and mother collide to produce a third worldline - a baby.
Worldlines are not just traced out by large objects. Rather, large objects are the collection of the many smaller worldlines of their elementary-particle constituents. By this view, humans are simply large numbers of particle worldlines temporarily traveling together. The particles making a up a human were scattered about before he was born and will re-scatter after his death. Thus, the worldline traced out by each one of us contains not only all the information of our entire history, but also that of each individual particle in our bodies, going all the way back to the Big Bang! This led famous Russian cosmologist George Gamow to publish an autobiography titled My World Line.
Certain solutions of Einstein's equations admit time travel; a rather unpleasant possibility to physicists because of the possibility of time paradoxes. In general, two types of time paradoxes exist: the type in which past events are altered, and the type in which an event has no past. The first type has already been discussed; it is impossible to alter the past because worldlines cannot be broken or made to materialize from nothing. The second type is much more complex, and worldlines alone apparently cannot resolve it. For example, suppose a young woman is working on a revolutionary new mode of hyperspace travel that would allow humanity to colonize other solar systems as well as travel to the past. She is mysteriously visited by an elderly woman who gives her the formula to create this new mode of travel. She immediately publishes her results and within ten years colonies have been set up on several other stars. She moves to a colony and lives happily, later traveling back in time to visit her younger self and fulfill her own past. If this is true - and, since worldlines can loop back on themselves, it is theoretically possible - then where did the original idea come from? The woman's worldline is a closed circle.
This represents one of several types of closed timelike curves, or CTCs. Such curves are among the possible solutions to the equations of general relativity. Mathematician Kurt Gödel, who had also proven that arithmetic is fundamentally inconsistent from a logical viewpoint, derived the first real mathematical solution to general relativity that permitted time travel. In Gödel's universe, following the path of a particle led one to come back around and meet oneself in the past. However, Gödel had made a problematic assumption: his solution depended on the slow rotation of gas and dust through the universe. Since experiments showed that gas and dust in the real universe are not rotating, Gödel's solution was dismissed.
Over the years, a number of other CTC solutions have been found. Ezra Newman, Theodore Unti, and Louis Tamburino created a solution that appeared to be a black hole, but allowed a drastic form of time travel - moving 360° around their black hole did not lead you back to your starting point, but to an entirely new region of time. This universe was dubbed the NUT universe after the initials of its creators. The Kerr solution, one of the best relativity-only descriptions of black holes available, can be shown to admit time travel if an object can withstand the gravity of a black hole and travel through the black hole's center.
Physicist Kip Thorne, at the request of Carl Sagan, decided to try creating a theoretical time machine with a wormhole that would be traversible to space travelers. His hypothetical wormhole is large enough to admit travelers, does not seem to cause time paradoxes (that is, it allows only fulfillment and not alteration of the past), and will not close up, rip travelers apart, or freeze them in time. However, they do rely on a new, never-observed type of exotic matter with negative energy. Thorne has also devised several ways to create his wormholes given a supply of exotic matter and a large amount of energy. However, testing his ideas is not technically feasible at present. Moreover, physicist Stephen Hawking (who ironically has his own theory of wormholes connecting many universes) disagrees with Thorne, saying that Thorne had not accounted for radiation inside such a wormhole.
In Higher-Dimensional Geometry, four-dimensional beings are discussed. If the additional dimension into which they extend is spatial, they will have the ability to walk through walls, appear and disappear, and move through solid barriers without breaking them. They could even intertwine two solid, separate rings, remove objects from sealed bottles, and a number of other feats impossible in three dimensions. 
If their extra dimension is time, they will see the sequence of events experienced in steadily progressing order by three-dimensional beings as an infinite series of superimposed images representing our motions through time. For example, where we would see a ball dropping, they would see a series of images representing the ball in various stages of dropping. (Part of such a sequence is given in the image at right.) Such sequences are actually the time-dimensional aspect of our worldlines, which are composed of all our motions through all four dimensions. Thus, the view from the fourth dimension is a view of worldlines. (These are technically worldsheets, since they are traced by multidimensional objects.)
Although a more string-centric view of this topic can be found in the String Theory series, it will be discussed briefly here. Two-dimensional strings trace out two-dimensional worldsheets. Since strings, according to Feynman's sum-over-paths formulation of quantum mechanics, simultaneously travel by all paths from one point to another, they are always passing by every point in space. According to physicist Edward Witten, this property of strings ensures that six-dimensional figures called Calabi-Yau spaces (theorized to be the shape of the other dimensions of our universe) can be transformed by certain topology-changing deformations called flop transitions without causing physical calamity. This is because strings are constantly sweeping out two-dimensional worldsheets that shield the flop transition point from the rest of the universe. A similar thought process goes toward the ability of Calabi-Yau spaces to undergo more drastic changes called space-tearing conifold transitions.
Created by Dan Corbett, Kate Stafford, and Patrick Wright for ThinkQuest.