The goal of this paper was to present the theory of divisors on algebraic curves, which was first introduced by Dedekind and Weber. Using divisors we expose some basic Riemann ideas in the theory of algebraic curves. The first chapter contains a brief overview of the classical algebraic geometry: definitions of varieties, mappings, tangent space and nonsingularity. In the second chapter we have defined regular differential forms on the general variety and described modules of differential forms in the neighbourhood of nonsingular points. We have shown analogous results for the regular differential p-forms. We defined rational differential p-forms and showed that they form a vector space over the field of rational functions. In the third chapter, we defined the divisor group on curves. We have shown that the degree of the principal divisor on nonsingular projective curve equals zero and using this result we have proved one special case of Bézout’s theorem. Furthermore, we have shown that the Riemann-Roch space of a divisor on a curve is finite-dimensional and we deduced a simple criterion for testing the rationality of a curve. We studied divisors on a nonsingular plane cubic curve and derived the characterization using the dimension of divisor. Finally, we calculated the genus of a plane curve depending on the degree of a curve and using the Riemann-Roch theorem we scetched an idea for the classification of curves.