Descriptive statistics describe a sample or a population. Mathematical statistics is a branch of mathematics involving the development of the theory of statistics. Left out of this dichotomy is inferential statistics, which is about inferring from a sample to a population.
What is a population? A population is the entire body of subjects you are interested in. For example, if you are studying the difference in IQ between men and women in the United States, then the population is all the men and women in the United States. If you are studying the relation between type of computer chip and processing speed, then the population would be all the computers in the world. Not all populations are so large: If you are trying to find out who is the most liked professor in a department, then the population is all the professors in that department.
What is a sample? A sample is a smaller group drawn from the population. In the first example it would be American men and women, in the second it would be computers, in the third it would be professors.
As mentioned above, descriptive statistics can be about either populations or samples. If the data are continuous (e.g. weight, height, IQ, income) then typical descriptive statistics include measures of central tendency and spread. Typical measures of central tendency are the mean and the median. Typical measures of spread are the standard deviation, the variance and the interquartile range. If the data are discrete (e.g. hair color, religion) then descriptive statistics usually consist of a frequency table, listing the frequency of each possibility (e.g. blond, brunette, red, silver, white, gray etc.).
Often, when we take a sample from a population, we are more interested in the whole population. Going from the sample to the population involves inferential statistics. There are a huge number of inferential methods (e.g. all forms of regression, t-tests, chi-square tests and many others). However, the general procedure is to first propose a null hypothesis (nearly always, this is something like “there is no difference” or “there is no relationship” or something similar) and then test it. We calculate an effect size and then compare that effect size to what we might expect in the sample if the null hypothesis were true. For example, if we sampled 40 men and 40 women and tested their IQ, we could compare the means using a t-test and then see how large the result is.
One of the most commonly abused ideas in inferential statistics is the p-value. The p-value is the probability that an effect size as large as the one we found, or larger, would occur in a sample the size of the one we have, if, in the population, the null hypothesis were true.
All of this is part of applied statistics or data analysis. Mathematical statistics is the field that develops the theory behind the above processes. Books on applied statistics and data analysis are full of examples. Books on mathematical statistics are full of theorems and proofs. People in many fields study data analysis and applied statistics. Mathematical statistics is studied almost exclusively by mathematicians. In fact, unless you are a mathematician, it is unlikely that you’ve ever seen any mathematical statistics.