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What is topology of real algebraic varieties?

This is not an easy question. The difficulty is that a reasonable answer must rely on some background, which varies. Indeed, the answer depends on the answer to the question:
"What is variety?"

1. For a simple-minded answer at the high school level, please, read

Easy reading on topology of real plane algebraic curves by V. Kharlamov and O. Viro.

This is an elementary introduction, which requires preliminary knowledge neither in topology, nor algebraic geometry, but can give the first impression of the subject. It contains almost no proofs, and should be considered a sort of pure advertisement.

This was written as an informal introduction to a book Topological properties of real plane algebraic curves by V. M. Kharlamov, V. A. Rokhlin and O. Y. Viro. The book has not been completed.

2. Another introduction to the same subject is given in the introductory part of my book Real algebraic varieties with prescribed topology,. It requires less topology, at least at the beginning. The book is unfinished.

3. A tremendous role in shaping of the subject was played by Hilbert's sixteenth problem. Take a look at transparencies of my talk The 16th Hilbert problem, a story of mystery, mistakes and solution, screen version.

There is also a complete text of Hilbert's talk at the second International Congress of Mathematicians in 1900 with all 23 problems.

7. Soon after the sixteenth Hilbert problem was formulated, a former Hilbert's student Virginia Ragsdale published a paper On the arrangement of the real branches of plane algebraic curves, Amer. J. Math. 28 (1906), 377-404, in which she analyzed the results by Harnack and Hilbert and posed several conjectures. Some of them were proved, the others disproved. A story about this is presented in the paper Patchworking algebraic curves disproves the Ragsdale conjecture, by I. Itenberg, O. Ya. Viro, Math. Intelligencer 18 (1996), no. 4, 19-28.