The sun, like all stars, progresses through its own life cycle of birth, adulthood, and old age. A star is born when a giant cloud of gas and dust collapses and begins to release energy due to this gravitational contraction. Eventually, the density within this proto-star becomes large enough that hydrogen begins to fuse into helium; once this process of nuclear fusion begins, a star is considered to be a main sequence star--where it will spend most of its adult life. Our sun has been a main sequence star for the past four and a half billion years and will continue to convert hydrogen into helium for about five billion more years into the future. The steady stream of light from an adult main sequence star provides a relatively consistent source of energy for planetary systems--and life--nearby.
That said, even main sequence stars change a little in brightness over their lifetime, which is enough to affect life. When Earth first formed, the sun was about 70% as bright as today, and at the end of its main sequence lifetime the sun is projected to be about 120% brighter than today. This gradual brightening occurs as more and more hydrogen atoms are converted into helium within the sun’s core--to maintain the same pressure, the remaining fewer molecules now move faster (i.e. temperature increases), which speeds up the rate of nuclear reactions. The result is that main sequence stars steadily increase their rate of burning over their adulthood. For life in the Solar System, this means that habitable zones will change with the aging of the sun.
Eventually the sun’s increasing brightness will cause Earth’s oceans to evaporate, making life as we know it impossible. This will occur somewhere between one and four billion years in the future--so perhaps we still have time to find technological solutions (such as extreme geoengineering, or perhaps even relocating our planet to a more distant orbit). Mars, on the other hand, will be comfortably within the habitable zone at this time, so perhaps future humans may establish colonies or even terraform the red planet as a new home. More distant locations in the Solar System may also be suitable for human settlement as the sun warms, including Titan and other satellites orbiting Jupiter and Saturn.
Because of its hazy atmosphere, Titan may provide a rather consistent climate as the sun slowly brightens. This is because Titan’s atmosphere provides both a warming greenhouse effect--because of the absorption and re-radiation of energy by methane and other gases--as well as a cooling anti-greenhouse effect. This anti-greenhouse effect is due to scattering of incoming sunlight by haze particles and other molecules in Titan’s upper atmosphere. These two processes somewhat compensate for each other, so that a brighter sun does not necessarily mean Titan’s surface will get much warmer. In fact, some calculations suggest that the anti-greenhouse effect may even increase as the sun brightens, which would cool Titan’s surface slightly. In any case, the competing effects of warming and cooling from Titan’s complex atmosphere will keep it a bit more consistent (although still much colder than Earth) during the sun’s adult life.
Eventually the sun will run low on hydrogen fuel in its core and will start to fuse the scant remaining hydrogen in an expansive shell surrounding the core. This marks the sun’s evolution into an elderly red giant phase as well as the physical expansion of the sun to engulf most of the inner solar system. At this time, about five to six billion years from now, the sun will swell up to swallow Mercury, Venus, and Earth. Mars will probably survive, but it will be far too hot to support liquid water on its surface. During the course of evolution into a red giant, the sun will reach a peak luminosity several thousand times that of today--so any remaining life in the Solar System will need to relocate or develop serious shielding technology in order to survive.
Even the atmosphere of Titan may not survive the drastic shift in brightness during the red giant phase of the sun. At its peak brightness, the Saturnian system will receive about twenty times the energy of present-day Earth, which could cause Titan’s atmosphere to turn into steam. That said, the transition period when the sun moves from a main sequence star to a red giant will provide about a hundred million years where Titan receives about half the amount of energy from the sun that Earth does today. This could be useful for future Titan colonists, who would previously have adjusted to life in a much colder planetary system.
Eventually the sun will dwindle into old age as a small, dense white dwarf star with no fuel left. The sun will gradually cool and dwindle into a brown dwarf and then finally into a cold black dwarf that emits no energy at all. The outermost planets will likely survive the sun’s evolutionary cycle, but life in the Solar System may not be as resilient. While life could conceivably reside on a planet orbiting a white dwarf, only a species with considerably advanced technology could withstand the expansion of the sun into a red giant.
Long-term survival of any species, past the lifetime of its parent star, will ultimately require interstellar travel and colonization. But for the near-term future, the outer Solar System may at least provide an intermediate place for future humans to relocate.