Subject probability is probability based on belief. You can have probability which is based on actual data (i.e. information you collect) or you can have it based on belief. The belief aspect varies: you can have mathematical assumptions and arguments to justify the distribution or it could be a gut instinct.
An example I will use is the Binomial distribution.
In the binomial distribution we assume the following:
1) We have n different events
2) Each event has a probability p of a success and 1 - p of a non-success.
3) Each trial is independent.
Based on these assumptions we do some math taking into account P(A and B) = P(A)P(B) for independence and with some combinatorics get the Binomial distribution.
If we think that the thing we are looking at follows these assumptions then we will assign a binomial distribution to it.
It may not follow the assumptions and be wrong, but our argument for using it is based on what we think is the best set of matching assumptions with respect to some model. If a model with certain assumptions doesn't exist, then a statistician will use theory to build such a model.
The other kind of subject probability is just gut instinct. You may just have a feeling that something has a certain distribution but you don't have as much justification like you did with the assumptions case of the Binomial. The binomial case was very specific, but the gut instinct is not as specific.