Symbols, Syntax, and Semantics
Symbols, syntax, and semantics are the building blocks of every language.
Symbols
In general, symbols are any entities that humans use to represent something else. Think of symbols as a type of shorthand; they aid in abstracting the essential properties of things in a particular context, so that people may think about and manipulate those things more easily in that context.
Consider a circle in the context of mathematics.
To avoid a potentially infinite regression or circular discussion over what symbolizes what, let the above circle visually symbolize the mathematical idea circle - a point together with all the points a given distance from it all in the same plane (note that even this definition is a kind of symbol). Circles have properties, and thinking in terms of their properties enables you to more easily manipulate and use circles. One of the properties of the above circle is that it is red. Of course, in most mathematical contexts this is an irrelevant property. On the other hand, in many mathematical contexts the ratio of a circle's circumference to its diameter is often a very useful piece of knowledge. When someone wants to think or write about this property they may say or write, "the ratio of a circle's circumference to its diameter." Although this string of words symbolizes the property, it is obviously not very efficient. Since we know that for all circles this ratio works out to the same number, how about using the number itself as a symbol for the ratio?
In fact, taken to extremes, the number approach is more inefficient than
the string of words approach. Because even a number like
3.1415926535897932384626433832795 is not long enough to represent exactly the ratio of any
circle's circumference to its diameter. The obvious conclusion of all this is that
the symbol for the Greek letter pi, 
,
is a mighty fine and efficient way to symbolize the idea of the ratio of a circle's
circumference to its diameter.
Syntax is the set of rules that enable you to distinguish, in a technical sense, what strings or collections of symbols constitute a valid sentence or expression. For example, your knowledge of English syntax tells you that the string of characters ;lik*789 is not a valid expression in English, whereas that same knowledge tells you that the sentence you are reading is a syntactically valid English sentence.
Semantics
Given a syntactically valid sentence, semantic rules tell you if the sentence has any meaning in the language. "Ran jump cow cow fish cat do not stupid my my hers of." is syntactically a nominally valid sentence in English. However, your innate knowledge of the semantics of English tells you in fact that it is not a valid sentence in the English language.
Often times a fine line exists between syntactic and semantic rules, and a
lot of circular reasoning can take place. For the most part though, the distinction
is usually clear.
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