Let $f \in \mathbb{C}[x_0, x_1, y_0, y_1]$ be a polynomial such that $f(\lambda x_0, \lambda x_1, y_0, y_1) = \lambda^af(x_0, x_1, y_0, y_1)$ and $f(x_0, x_1, \mu y_0, \mu y_1) = \mu^bf(x_0, x_1, y_0, y_1)$, then $$X = \{([x_0, x_1], [y_0, y_1]) \in \mathbb{CP}^1\times\mathbb{CP}^1 \mid f(x_0, x_1, y_0, y_1) = 0\}$$ is a curve. If $X$ is smooth, it has genus $(a-1)(b-1)$, so every genus can be realised. As $\mathbb{CP}^1\times\mathbb{CP}^1$ embeds in $\mathbb{CP}^3$ via the Segre embedding, $X$ is a curve in $\mathbb{CP}^3$.
Another way of constructing curves in a projective space is via complete intersections. Let $f_1, \dots, f_{n-1} \in \mathbb{C}[x_0, \dots, x_n]$ be homogeneous polynomials of degrees $d_1, \dots, d_{n-1}$ respectively, then
$$Y = \{[x_0, \dots, x_n] \in \mathbb{CP}^n \mid f_1(x_0, \dots, x_n) = \dots = f_{n-1}(x_0, \dots, x_n) = 0\}$$
is a curve. If $Y$ is smooth, it has genus $1 - \frac{1}{2}(n + 1 - d_1 - \dots - d_{n-1})d_1\dots d_{n-1}$. This construction gives rise to many genera that don't appear in the degree-genus formula, but not all of them: see this sequence. For example, there is no choice of dimension $n$ and degrees $d_1, \dots, d_{n-1}$ which give rise to a genus two curve, i.e. a genus two curve is not a complete intersection.
As soon as the degree is larger than two, there are non-smooth irreducible curves of degree $d$. Take for instance $z^{d} - t w^{d-1}=0$ in the projective plane, in the homogeneous coordinates $[z:w:t]$. It has a cusp at $[0:0:1]$.
Addendum: in fact, there is a whole (difficult) field in algebraic geometry whose goal is to understand singularities of irreducible algebraic sets. Topics include classification and finding desingularizations (roughly speaking, finding a way to "fix" the singularity). Those questions can also be studied over different fields.
Best Answer
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If $V$ is a curve, then $V$ is a closed set in Zariski topology $\Bbb A^n_k$. The following are equivalent:
$(1)$ V is irreducible
$(2)$ any two non-empty open sets of $V$ have a non-empty intersection
$(3)$ Every non-empty open subset is dense in $V$
$(4)$ Every non-empty open subset is connected
$(5)$ Every non-empty open subset is irreducible