The sun will eventually become a red giant; it may or may not envelop the Earth - if it does, the Earth will be vaporized. Long before it does, however, the heat from the luminous sun will cook the planet until it is sterile, eventually melting it. This will happen in about four billion years!

Long before that, however, the sun will become bright enough that while it does not melt the planet, it will cause climatic effects that will make life impossible. We do not know how the climate will respond to such increased solar input, so we do not know how much the climate can take before the sun cooks us.

Our climatological models aren't that good. Most climatologists seem to think that it will be a few billion years.

The question of when the world will end is one that can be answered on many levels. Since I am a geologist, I am going to interpret your question to mean, "When will the actual planet Earth cease to be a planet?"

There are two major methods by which our planet can be destroyed;
Evolution of the sun into a red giant star
or impact with a giant extra-terrestrial body.

The sun is a star that is in the middle of its life. In about 6 billion years the sun will have exhausted its hydrogen in its core and will cease to create helium by fusion. This will cause the sun to cool enough that it will collapse under its own gravity. As the sun collapses, the atoms inside of the sun will be pulled closer together and will collide with each other more frequently, creating more heat and raising the temperature once again. Eventually it will become hot enough to fuse heavier elements together. Fusion of heavier elements will release enough energy that the sun will become a red giant. When the sun becomes a red giant its radius can expand to be ~100 times larger than it currently is. This will engulf Earth with temperatures much higher than the melting point of the rocks Earth is made of destroying the planet.

Early in Earth's history (about 4.5 billion years ago), a smaller (~70% the current mass) Earth was struck by a Mars-sized impactor (bolide). The impactor struck the earth at an angle and delivered enough energy that it melted the planet and itself actually vaporizing some of the rocks that make up the crust. The energy from the angled impact spun some of the amalgamated body off the earth which collected, coalesced, and cooled to form the Earth's moon. An impact of this size would definitely rework the planet effectively ending this earth.

I would not worry about either of these events.

The probability of a giant impact is incredibly low and 6 billion years is a really long-time. There are plenty of other ways that human presence on this planet can end like global warming and other environmental catastrophes, but the planet will persevere.

In about 5 billion years, when the Sun becomes a red giant, and gets as big as the orbit of Mars!