This was originally my homework assignment. I thought it would make a good addition to this node.
Note: An mb is a millibar (a unit of pressure) and a K is a Kelvin
Terraforming, being a sci-fi concept, has hundreds of approaches. Some are feasible, while others are pure fantasy. Which methods will be used will depend on the money available and how quickly the people want it completed.
There are a myriad of ways to heat Mars so that liquid water can exist on its surface. Most methods raise the polar temperature by 4 Kelvins (K) to sublime the southern ice cap, releasing around 50-100 mb of carbon dioxide into the atmosphere. This would raise the mean temperature by 10 K, which would start to release more carbon dioxide trapped in the regolith. As more carbon dioxide is released, more heat is trapped by it, which causes more carbon dioxide to be released. This process is called positive-feedback greenhousing. For liquid water to exist at least in the equatorial regions, around 600 mb of carbon dioxide needs to be sublimed. It is not known exactly how much carbon dioxide exists on Mars, but estimates range from 100-2000 mb.
The southern ice cap, being white, reflects much of the sun's energy. If dark-colored dust were spread over the cap, more of the sun's energy would be trapped, allowing the cap to sublime and start the positive feedback process. One way to spread the cap with dust is to detonate 20-KT nuclear weapons, underground, in a dust drift located near the south pole. The amount of dust created in the nuclear detonation would be enough to cover the whole cap, subliming it. The ideal detonation time would be during the commencement of spring in the southern hemisphere of Mars. A new bomb will need to be detonated every year at the beginning of spring until all of the cap has sublimed, which is estimated to be four martian years (7 earth years). This is by far the cheapest, easiest, fastest, most automatic and most feasible method. We could start this today if we wanted to. However, the social hatred (unfounded, IMHO) of nuclear devices will probably prevent this method from being used.
A class of primitive bacteria called Methanogens can survive on today's Mars, and would create methane, a greenhouse gas, from carbon dioxide and hydrogen contained in the regolith. This method is also very cheap, easy, and automatic, but it is not very feasible, as the methane would last mere hours before being broken up by the sun's powerful ultraviolet rays.
Perfluorocarbons are very powerful, long-lived greenhouse gases. Perfluoromethane (CF4) is an example. Perfluoromethane is 10,000 times more effective a greenhouse gas than carbon dioxide, and lasts centuries even in Mars's tenuous atmosphere. It is made by releasing pure fluorine into the Martian atmosphere. The fluorine reacts with carbon dioxide to form perfluoromethane and oxygen. Automatic factories could be inserted on the martian surface that would mine apatite for fluorine, using solar power. The main advantage of using PFC's is that it is the most feasible method that is socially acceptable. It is also very flexible; time needed is directly proportional to cost.
A soletta is a fancy name for an orbiting space mirror. To be able to sublime the southern cap, the soletta would need to be about 130 km in diameter, and would weigh around 200,000 tons. Such a mirror would be too big to launch directly from Earth, but could be contructed in space using materials from asteroids. The mirror would be placed 214,000 km behind Mars, allowing it to hover as a statite; the light pressure hitting it balances the gravity pulling it in. Solettas have the added advantage of flexibility; solettas can also melt permafrost and release volatiles contained in the regolith. However, the construction of a space mirror would be a overly grandiose project, taking several years and costing billions of dollars.
Moholes are simply very deep man-made holes that release geothermal energy. Moholes would not be expensive, and the idea is not far-fetched. However, whether this idea would actually work is still being debated.
Another speculative method is to put electric heaters on Mars powered by wind turbines. As with moholes, many see this method as highly futile because the radiative heat created would just escape into space.
Many scientists have predicted that asteroids made out of solid ammonia exist in the outer solar system. Nuclear rocket packs would attach themselves to the asteroid, using some of the ammonia as propellant, and utilizing gravity assists. The further the asteroid is from Mars, the less energy is required, but the longer it will take for the asteroid to reach Mars. When the asteroid is in Mars's atmosphere, explosives (probably nuclear) would be used to break the asteroids into many small meteorites. This would release ammonia, a greenhouse gas, and would melt some permafrost. Ammonia, like methane, is a short-lived greenhouse gas, and would only last for hours. However, bacteria could be used to "recycle" the disassociated ammonia. Asteroidal impacts, however, will dramatically change Mars's landforms. This will anger some people. Dust created is also a negative.
If regolithal carbon dioxide reserves prove to be too small, then more heating will be required. PFC production would be the only effective method in this case. Heating beyond the 4 K needed to sublime the southern cap is recommended anyway, as added heat will melt the northern ice cap, a vast source of water.
Preparing Mars for Plant Life
Now that Mars has a thick atmosphere of carbon dioxide and a much warmer climate, only water and nitrogen will need to be added for plants to thrive on Mars. Waiting for the permafrost to melt would take centuries. Melting the northern ice cap, which is water ice, would be much quicker. Nitrogen is thought to exist as nitrates in the regolith. However, this has not been proven.
The easiest way to melt the northern cap is to pump enough PFC's into the atmosphere so that the average temperatures of the polar regions rise above freezing. This would make the tropics uncomfortably warm, however. Using PFC's is much cheaper and easier than using solettas.
Solettas, described in the previous section, could also be utilized to melt the ice cap. This will probably be used in conjunction with PFC's. Solettas could also be used to release nitrogen from nitrates in the soil.
Nanotechnology could also be used to melt ice and release nitrogen. Self-replicating "nanites" would be smaller than a bacterium, yet they are able to work very quickly, allowing total terraformation in a matter of decades. Nanotechnology, however, is an invention of the future.
Asteroidal impacts could also release nitrogen and/or water, but this would be a waste of the asteroid's energy, and this would leave huge craters, and fill the atmosphere with dust.
Oxygenating the Planet
Although a carbon dioxide atmosphere would be fine for plants, animal life requires an oxygen-containing atmosphere. Plants take in carbon dioxide, and release sugar and oxygen. This will be the predominant oxygenation method, but will probably take longer than a century. Black chlorophyll would shorten the time needed considerably.
The Bosch process could supplement plants in oxygenation. In this process, carbon dioxide and hydrogen combine to form water and carbon. The water is electrolyzed to form hydrogen and oxygen. The oxygen is pumped into the atmosphere, while the hydrogen is put back into the process. Power for this process will need to come from wind or nuclear power, as solar power would be better utilized by plants.
The terraformation process could take as little as twenty years or as much as 100,000 years. My opinion is that it will take around a century or two. However, considering it took billions of years for the same processes to occur on earth, total terraformation will be a quick process, no matter how it is reached.