There are a lot of factors that contribute to the bitterness of coffee. Some of these have to do with the concentration of various compounds in the coffee and the roasting process, but the way you brew the coffee can make a big difference as well.
Assuming that you are not going to be harvesting and roasting your own coffee, we can focus on the brewing process.
Buying Beans: Robusta vs Arabica
To decrease perceived bitterness in coffee, the best choice is Arabica. Arabica beans are often sweeter with notes of sugars and fruits and a somewhat wine-like acidity. Robusta on the other hand tends to be harsher with more bitterness. This difference in quality is reflected in cost however, as Arabica tends to be more expensive than Robusta.
In terms of roast level, medium roasts generally contain fewer soluble solids, more acidity and a stronger aroma than dark roasts. All of these should contribute to making medium roasts on average less bitter than dark roasts.
Water: Temperature and Chemistry
You have to find the sweet spot when it comes to temperature, a little trial and error can be used here depending on the blend and brewing method. For example, the people at AeroPress recommend a much lower temperature to reduce bitterness that what you would use in a french press.
In general research has shown that coffee brewed in hot water is perceived to be less bitter than when cold water is used. Additionally, bitterness is reduced when hard or soft water is used when compared to distilled water.
Bitterness is strongly correlated with the amount of dissolved solids in the brewed coffee. To ensure that the amount of dissolved solids and the overall extraction is correct (and minimally bitter), make sure to use the correct grind coarseness, water temperature, and brew time for your chosen method. Over extraction can be a major cause of bitterness.
Drip and pour-over brewing methods tend to have less bitterness than a french press or other immersion methods. This can be attributed to less dissolved solids in the coffee.
Much of this section will be outside the control of most, but it is interesting nonetheless.
Research has found that the introduction of polyphenols reduces the bitterness of coffee. The metallic bitterness found in some coffee has been attributed to dicaffeoylquinic acids. Trigonelline is perceived as bitter at concentrations of 0.25% and its degredation is proportional to roast level. Roasting Trigonelline produces a byproduct called pyridines and helps create the roasty aroma found in some coffee.
Chlorogenic acid is also found in coffee, higher concentrations in Robusta than Arabica, and that contributes to the bitterness as well. Additionally, Quinic acid is a product formed from the degradation of Chlorogenic acid and is present in coffee at twenty times its taste threshold, contributing to the bitterness.
Furfuryl alcohol is also known to increase the bitterness of coffee.
Coffee Chemistry: Cause of Bitter Coffee
Coffee Basics: The Difference Between Arabica and Robusta