1) I want to learn about algebraic curves and i'm confused, please correct me if i'm wrong : when we say an Affine algebraic curve over the field $F$ :
here affine to distinguish it from projective and $F$ is the field of coefficients and zeros of the polynomials $P_i$ in the equations $P_i=0$ defining the algebraic curve. Being a curve, we have $n$ variables $x_1,...,x_n\in F$ and $n-1$ equations $P_i(x_1,...,x_n)=0$. When we have only one equation and 2 variables $x_1,x_2\in F$ we can add the word "plane curve" to the name of the curve. When $F=\mathbb R$ we can add the word "Real curve" instead of saying over the field $F$.
2) every curve is at the same time affine and projective: if we have an affine curve we can turn it into projective curve by a sort of change of variables and a multiplication to put the polynomials into homogeneous form.
3) In wikipedia http://en.wikipedia.org/wiki/Algebraic_curve we read " An algebraic curve likewise has topological dimension two; in other words, it is a surface. " what is a topological dimension here and what is a surface here? the line and the circle are algebraic curves but they are not surfaces!!!!