Terraforming means changing something to be earthlike. The sense in terraforming a planet is to making it more habitable for humans. If the cost of terraforming Mars is below the price of the whole surface of Mars, terraforming is worth the cost. 

We know that Mars is the fourth planet in the solar system, Earth the third. Because of that, Mars is colder than Earth. It is colder and drier than any place on Earth. It also only has a very thin, poisonous for humans atmosphere. Above that, the surface has some peroxides that would destroy any known living tissue. 
But it is still the planet that looks most like Earth in the solar system ( Earth excepted ). The day-night cycle is almost the same as Earth's, which allows plant growth in greenhouses. 

When we compare our moon to Mars, we see that Mars is much easier to terraform then the Moon. The moon's 28 day day-night cycle is unsupportable for plants and isn't comfortable for humans either. Plant growth on a greenhouse on the moon wouldn't be possible either: solar radiation would destroy them and if the greenhouse was made to shield the solar radiation ( a very thick greenhouse ), the temperature differences would kill the plants: Much too hot during day-time, much too cold during the night. 
Eventhough the moon has large amounts of oxygen, it is bound to the moons soil in very hard to separate forms like SiO₂ or MgO. 
The moon has only hydrogen in very small amounts in form of water in craters near the pole, where sunlight never reaches the ice. 

The only advantage of the moon as opposed to Mars is that He3 ( helium3: 2 protons, 1 neutron. The usual form of helium is He4: 2p⁺, 2n⁰ ) is found in it's soil. He3 is as good as nonexistent on Earth and no depots are known on Mars. This gas is now used in medicine to make very high-quality photos of inner organs of the human being, especially the lungs. It will also be used in second generation fusion-energy-plants or fusion engines in rockets. 
But nowadays, there is no big need in He3, so we don't need any expensive colony on the moon. 
 

We know that Mars, too, has large quantities of oxygen, but on Mars, the oxygen is bound in looser forms than on the Moon. The oxygen in mostly bound in form of carbon dioxide in the southern, some in the northern polar cap and in the regolith ( regolith is another word for soil on an extraterrestrial object ). Oxygen is also found in the toxic peroxides ( like hydrogen peroxide (H₂O₂) ), mostly on the upper layer of the regolith. 

Peroxides give of oxygen when heat is applied to them. The carbon dioxide, when solid, sublimates to become carbon dioxide gas ( carbon dioxide needs high pressures to be found in liquid form ). The oxygen from the peroxides can be breathed, the carbon dioxide cannot. Plants are exactly what we need: On earth, all plants suffer from carbon dioxide lack. Through photosynthesis, plants produce oxygen from carbon dioxide. 

Carbon dioxide has the positive feature of being a greenhouse gas. On earth, global warming is maybe due to a slightly higher concentration than it used to be. Venus also is so unsupportably hot ( almost 500'C day and night ) because of this gas. If Venus had Earth's atmosphere, the average temperature should be around 25'C ( Earth; 15'C ). 

Mars also has water in frozen form. The moon has very little, certainly not enough to cover a big part of the surface with liquid water. Mars has the advantage of having some in the northern polar cap. Larger reservoirs might be found underground in frozen, and maybe even, thanks to salts or areothermal heat, in liquid form. All in all, there might be enough water on Mars to have a big ocean on the northern hemisphere and a sea in Hellas Planitia, the largest crater in the solar system.  

If Mars shouldn't have big water reservoirs, but which it very well might have, the possibility of importing water from elsewhere is still there. Comets are made of water and can be brought to Mars with less energy than expected. Other chemicals like the greenhouse gas ammonia could be imported that way, too. (See: Moving Ammonia Asteroids). 

The idea in terraforming Mars is the following:  
Melting the southern polar icecap to release large quantities of carbon dioxide into the atmosphere, which, because of the greenhouse effect of this gas, will heat up the planet, maybe enough to release more carbon dioxide from the regolith and maybe even melting the northern polar icecap (Underground waterdepots would take a very long time to evaporate to get onto the surface of Mars) without extra heating. 
When the step of enriching the atmosphere with carbon dioxide (and nitrogen) and the surface with water is achieved, the second step starts: Enriching the atmosphere in oxygen (and nitrogen). 
This is simply done by releasing plants that can survive on Mars' still rude climate. Some genetical engineering might be necessary to achieve this step. 

The melting of the southern polar icecap is not as difficult as it might seem at first. A temperature change of only 4'C is required (there are several methods to reach such a temperature change as you will see later). If the temperature rise reached is not high enough for the regolith to exhaust its carbon dioxide, extra heating is necessary. There are several ways of doing this, too, as you will see on the second part of the terraforming site.  
 
 
 

Goal	Method	Result
Having a planet with temperatures and atmospheric pressures high enough to support liquid water at some places.	Melting of the south polar icecap.	A carbon dioxide rich atmosphere. Pressure between 50-150 mbars, temperatures a few centigrades higher
Having a protection against radiation.	Carbon dioxide enrichment of the atmosphere. Some oxygen from peroxides or gases like ammonia would help.	An ozone layer thick enough for good protection (solar radiation weaker at Mars than on Earth, carbon dioxide promotes ozone construction).
Having a planet with conditions able to support flora for food and oxygen enrichment of the atmosphere.	Carbon dioxide enrichment of the atmosphere, liquid water on the surface of the planet. Extra heating is probably required.	A warm and humid climate, in which plants can grow food for humans to eat. No spacesuit is required anymore, an oxygen masks is enough.
Having a planet with conditions for humans to survive on the surface with no oxygen masks.	Having a high population of plants that produce oxygen. This process will take a lot of time.	An earthlike planet that can be the home of a big part of the human population.