The short answer is that Classical logic is about the idea that complete sentences can only take one of two Truth values: every grammatically well-formed sentence in a classical logic is either True or it is False. The longer answer... takes a while.
The modern standard analysis of what Logic is can be described as its providing a framework with three different elements.
- A Logic has a Calculus; a formal system of symbolic terms that are composed to form complex strings. Some of these terms will use logical symbols such as ^, v, -> etc., but from the perspective of the Calculus, these things are just marks on a page or a computer screen. The formation of these terms is usually governed by rules of grammar, and we call strings formed in accordance with the rules concerning terms and grammar Well Formed Formulae.
- A Logic has a Deductive system. As a computer scientist, you might be familiar with this aspect of logic more than the others. A deductive system is a mechanism of rules of inference (and occasionally some implicit formulae, understood as Axioms) by which we can, when presented with certain formulae as input, do some simple computational work and determine that other formulae follow as output given the inference rules and the input. We call the relationship between the formulae input and output in such systems Syntactic Entailment, and it is often given the formal expression:
PREMISES |- conclusion
- A Logic has a Semantics; a mathematical picture of some background structure that we take things to have in order to talk about them logically. This might, for instance, just be talking about propositional Truth Values - some of the well formed formulae we have are True sentences, and some of them are False sentences. We would say {T,F}, the values that our logic can deal with, form the Domain of a Model in our semantics, and that we assign our Formulae, the things we want to manipulate in our logic, to these values using the Interpretation Function of a Model. If a sentence is interpreted as True in a model it is often written as:
MODEL |= true_proposition
(You'll also sometimes see that symbol used in sentences of the form "PREMISES |= conclusion"; this is shorthand for the relation that "all models that make PREMISES true also make conclusion true")
What makes Logic interesting is that deductive systems and semantics can be given rigorous treatments that tell us about what we can do with formulae. For instance, in Classical Propositional Logic, True or False are understood as forming a two-valued Boolean Algebra - a mathematical structure described in, for example, set theory or category theory. This means we can talk about particular kinds of mathematical function that we know are described in that algebra - if you have a true sentence A and a false sentence B, then according to the algebra, we can evaluate A ^ B as a false sentence (interpreting ^ as a function that takes T and F to F), and A v B as a true sentence (interpreting v to take T and F to T).
So in Classical Propositional Logic, rather than having to go through and manually specify values for every single formula, you can instead require that the interpretations assign values to all of the Basic formulae (like the atomic propositions) and build up values for more complex sentences inductively. This structure means you can have a theory of deductive inference that "gets things right" on the level of interpreting sentences in line with their mathematical relationships.
It is analogously possible to build logics over complicated logical calculi using more detailed mathematical models, such that we get very useful and productive deductive systems. In particular, the work in the late 19th-early 20th century by logicians such as Frege, Russell/Whitehead and Tarski gave rise to Predicate Calculus, which allows us to talk about Names, Functions, Properties and Relations of objects, and about variables that range over these things, in more detailed and general ways than just thinking of assigning truth values to sentences.
What is generally now called Classical logic is the framework that is thought to encapsulate both the Predicate calculus formalism and the compositional aspect that a structured, two-valued algebra of logic gave us in the propositional case. Predicate logic works with the idea that our domain of objects is something that we refer to in the logical structure of making assertions - when I say that "That chair is green", the logical form of what I'm saying actually identifies something in the domain to be "that chair", and to attribute to it the property of "being green". The logical structure of language is such that what I say is true when it turns out that "being green" is, according to the interpretation we make, something that the object that is "that chair" satisfies. We can introduce first-order Variables and Quantifiers to make generalizations about whether properties apply to some object, every object, no object etc. And in the Classical framework, every property is understood to either determinately apply to a given object or to determinately fail to apply to a given object; the payoff being that we can use another (more complex) algebraic analysis to let us build up expressive and complete deductive systems for our logic.
The power of deductive systems and the simplicity of the notion of ontology of this kind of framework is what makes it so widespread in the methodology of analytical philosophy, mathematics and science. It is by no means the only, or even necessarily the most appropriate, logic for use in formulating languages about particular subjects, but it is certainly one that has become well established in a great many areas of study.