The mixing ratio is a concept in atmospheric science that describes the ratio of wet to dry particles in the air. Conceptually it is similar to humidity but can refer to other elements as well. A mixing ratio can be calculated for atmospheric elements such as argon or ozone in addition to water. Small changes in elemental mixing ratios have important effects on the earth's atmosphere. The mixing ratio is calculated in different ways depending on what information you already have. The mixing ratio can always be determined from the humidity and the saturated mixing ratio.
Write down the humidity and the saturated mixing ratio. The humidity is a term you frequently hear in weather channels and determines the level of moisture in the air. The saturated mixing ratio is the maximum amount of water vapor that can be held in the air at a particular temperature. For an example let's say the humidity is 0.8 and the saturated mixing ratio is 0.5.
Multiply the mixing ratio by the saturated mixing ratio. In our example the answer is 0.8 x 0.5 = 0.04.
Divide the answer from the previous step by 100. In our example the answer is 0.04/100 = 0.004. This answer is your mixing ratio.
Double check your answer by dividing the mixing ratio by the saturated mixing ratio and multiplying by 100. The answer should be the same as the humidity; if it is -- you know you have the correct answer.
TL;DR (Too Long; Didn't Read)
The mixing ratio is calculated in different ways with different types of information. With other starting information you will have to perform additional calculation to find the saturated mixing ratio. The final units for the mixing ratio depend on what units the relative humidity is expressed in. These units are usually grams per cubic meter, but any density metric could be used.