There are a few possible ways to thicken a sauce, soup, or stock each having their own advantages and disadvantages.

One way is to make a roux. A roux is a blend of flour and a fat that is heated up and added to the liquid. Clarified butter is usually the most suggested type of fat used, but regular butter, oils, and rendered animal fats may be used. The mixture of 3 parts flour, 2 parts fat (many other books suggest equal ratios) should be heated in a suate pan to the desired color. A white or lightly cooked roux will thicken the sauce more then a darker roux, but will leave a more floury flavor. Cooking the roux darker will add a more nutty aroma and flavor, and can change the color of the sauce. A white roux would be better used in a sauce like bechemel or veloute. A brown roux which is cooked until it turns dark is commonly used in Cajun/creole cooking in dishes like gumbo.
The flour used also can impact how the roux will thicken, cake flour (higher starch to protein) will thicken more than bread flour. All-purpose flour is a mixture of the two and is the most commonly used. It is suggested to add a hot roux to a cold sauce, or to add a hot sauce to a cold roux (to avoid lumping). Avoid adding hot roux to a hot liquid, since it may splash and cause burns. Also, when adding roux the sauce will need to be stirred until thickened. Roux can be prepared ahead of time and stored in the freezer, freezing the roux in ice cube trays may help measure for consistency. 2 ice cubes with one tablespoon each of roux will thicken one cup of liquid. After the roux is added and the sauce is heated and thickened it is suggested to skim the sauce to remove impurities in the flour.
If fat floats on top of the flour, the roux has separated, throw out the roux and start over.

A beurre manie is mixture of equal parts flour and butter, kneaded together, so as the butter melts it mixes with the flour, and slowly thickens the sauce as you add more.

Another way to thicken a sauce or soup is to create a starch slurry. Corn starch is the most commonly used, but arrowroot, tapioca, rice flour, and potato starch can also be used. Corn starch is translucent, thickens when heated, but can gel when cooled, too much heat will reduce its thickening abilities. Arrowroot has the same thickening power as cornstarch, but will not gel when cooled, also will hold up better to heat for a longer period of time. Tapioca flour stronger then cornstarch, available at Asian grocery stores. Potato starch is the strongest of the starches, but costs more. Rice flour is the weakest of the starches, can be frozen, but can be expensive. Disadvantages also include that starches do not thicken with acidic sauces and a skin will form over chilled dairy sauces.

Liaison is a mixture of egg yolks and cream used to slightly thicken sauces and add a smooth texture, bold flavor, and a slight ivory color. Cream and eggs (8 oz cream to three egg yolks) are whisked together, this raises the cooking temperature of the eggs. After mixing the eggs and cream gradually add the hot sauce to the egg mixture while constantly. This process is called tempering and is done to prevent the egg from cooking. When about 1/3 of the sauce is mixed in the egg bowl you can dump the egg/sauce mixture into the pot with the rest of the sauce and whisk. Be careful not to let the sauce get over 185F or the eggs will curdle ruining the sauce. Flour in the sauce can help reduce the eggs from cooking in the sauce also. If the sauce gets to hot you can try to save it by placing the put in an ice bath to quickly bring the temperature down, whisk in more cream, you can also try to straining the sauce. Sauces in aluminum pots containing egg yolks can turn grey.

If the sauce is chunky try pureeing the sauce to thicken it.

File powder will also help thicken a gumbo, but should be added after the gumbo is removed from heat, and does not reheat well, so should be added right before gumbo is served.

Before adding a roux or starch to a stock always skim the fat off the top, once the roux or starch bounds to the fat in the stock it will be impossible to skim the fat out.

Gums like guar, xanthan, tara, carob, tragacanth, and arabic gums can also be used. Guar gum (E-412) is a powder made from the seeds of the leguminous plant. It is soluble in cold water and still works as a thickener under the use of high heat. Xanthan gum (E-415) is a powder produced from fermented cornstarch with bacteria from cabbage. It can resist thawing and freezing, is soluable in hot or cold water, stable in acids, and gels when mixed with equal parts carob gum. Carob gum (E-410) main use is to add elasticity to gels made with agar and carrageenans to help them avoid cracking. Tragacanth gum (E-413) is a thickener that works with acidic foods. Arabic gum (E-414) also called acacia gum or Senegal gum, is a weakest of the gum thickeners but can also be used to help emulsify essential oils into soft drinks.
The (E-NUMBER) is the European code for the elements found in food.