Subjective probability is one of three main intrepretations of the concept of probability (with classical probablity and statistical probability being the other two). In subjective probability, probabilities are seen as a degree of confidence that we may have that a certain event will occur. These probabilities are estimated using subjective factors, which may include personal judgement, intuition, expertise, and the like. In order to combine these factors in a meaningful way, Bayes' theorem might be used.
As an example of subjective probability, if the weather forecast predicts an 80% chance of rain, this prediction is based on a wide array of information about weather patterns and what has happened in the past when conditions were similar. Here is another example: If you take out an auto insurance policy on your car, the insurer may be interested in determining the probability that you will get in an accident for the purpose of calculating your premium. The auto insurer will likely have a large amount of information about you and about similar drivers to draw on in order to make an estimate of that probability, and hence your premium.
Unlike classical probability and statistical probability, subjective probability is used to describe unique events. Taking the example of the weather forecast, there is no sample space of all possible weather outcomes that are all equally likely, and the forecaster does not have access to thousands of identical universes to determine in what percentage of them it rains at a certain location. The forecaster must determine the percent chance of rain based on his knowledge of weather patterns and past weather events and his expertise and personal judgement. Of course, one person's judgement may be different from another's, so they might assign different probabilities to the exact same event. For example, two auto insurers may use a different set of factors to determine the probability that you will get in an accident, so their assessment of your risk, and hence your premium, may differ depending on what factors are taken into consideration.