What methods can I use to purify water? And what's the difference between those several methods (safety, duration needed, ...)?
Boiling- 185° water will become sterile in minutes. Bringing the water up to boiling point will typically sterilize it. At high altitude you may need to boil for a minute.
Chemical- Iodine, chlorine, and other chemicals can sterilize water. Follow directions on products specifically designed for this purpose. Typically you'll need to mix it and then let it sit for some number of minutes. Although chemicals are required, this method is often the most convenient as carrying a few pills is simple. If chlorine scares you, consider that you have been drinking it your whole life in tap water. Some battery operated systems work with salt and electrolysis, producing chlorine. This is convenient as well due to the availability of salt.
Distillation- This method works by distilling the water, usually in a solar still, to remove almost all impurities.
UV Light- Battery powered SteriPens produce UV light and kill everything including viruses. This is similar to some of the best pool water treatment systems, is very convenient and may work in less than a minute, although pens are costly and you'll need fresh batteries.
Filtration- As simple and quick as pumping the water through a filter. Each filter is different, but typically they remove 99%+ of bacteria, but not viruses unless a chemical is contained in the filter. You'll need to replace parts periodically and keep the system clean, but overall its pretty convenient.
Sunlight Pasteurization- Depending on the sunlight strength and outside temperature, you can leave a plastic bottle of water in sunlight for 6 hours+. The heat and UV light will make water safe. The downside is that you need to look up or guess how long based on the conditions.
Backwoods tricks- Survival techniques include many ways of obtaining water that is probably pure. They include distillation techniques (covered pits with pots, etc), and careful rainwater and dew collection or extraction and filtration by plants or clothing. None of these methods are convenient for daily use, and some are risky.
None of these methods necessarily protect against chemical contaminants, although some water filters are specifically designed to "filter" chlorine or other common chemicals. I have heard skepticism regarding some of those claims.
THE official information can be found on the CDC webpage.
The CDC defines the problem in terms of what you're trying to kill: Protozoa, Bacteria, or Viruses. They address each of those in terms of water treatment methods: Boiling, Filtering, Chemical Disinfection, (or now-a-days UV Treatment).
Please read what they have to say before you head into the wilderness. It is short and authoritative.
What does each method kill?
Here is my personal experience with each method: