The Library of Congress system has 21 broad categories, each represented by one letter of the alphabet. For example, books on philosophy, psychology and religion all are have codes starting with the letter B. Books on education start with the letter L. History of the Americas gets two letters: E and F. Many classes have subclasses identified by a second letter; for example, books on the history of education start with LA. Other classes begin with a single letter and then numbers (classes E and F, for example). Subsequent numbers further divide the categories.
The Dewey Decimal System first classifies books into nine broad categories, each identified by a number in the hundreds. For example, books on religion start are in the 200s; books on social sciences are in the 300s. Numbers are assigned within those categories; for example, 201 is for books on the philosophy of Christianity. Some topics use the 10s column for subdivision; for example, books on law are in the 340s; books on criminal law start with 345. Further division of categories is done by adding decimal numbers.
The Library of Congress system is an outline, much like one you would use to organize an essay. Subtopics are dealt with by indentation. The Dewey Decimal system is a hierarchy: Subtopics are dealt with by using hundreds, then tens, then units and then decimal places.
The Dewey Decimal System is used by more libraries, but the Library of Congress system is used by some of the biggest libraries -- most obviously the Library of Congress itself. Many universities use the Library of Congress system, but public libraries mostly use the Dewey Decimal System.