How is the period of an elliptic curve defined exactly? I sometimes read $\int_{E(\mathbf{R})} \frac{dx}{2y + a_1x + a_3}$ and sometimes $\int_{E(\mathbf{R})} |\frac{dx}{2y + a_1x + a_3}|$. Furthermore, one has to choose an orientation on $E(\mathbf{R})$.
So what's the correct definition for the constant appearing in the BSD conjecture?
 A: The comments above give already the answer, but for the sake of completeness let us be a bit more precise.
Let $E/\mathbb{Q}$ be an elliptic curve. Let $\Omega^{+}$ be the smallest positive element in the period lattice $\Lambda$. Then the conjecture of Birch and Swinnerton-Dyer predicts that
$$
\frac{L^{*}(E,1)}{[E(\mathbb{R}):E(\mathbb{R})^{o}] \cdot \Omega^{+}} = \frac{\prod_{p} c_p \cdot | Sha| \cdot Reg}{| E(\mathbb{Q})_{tors}|^2}
$$
The denominator on the left, where the index is the number of connected components of $E(\mathbb{R})$, can also be written as the absolute value of $\int_{E(\mathbb{R})}\omega_E$ where $\omega_E$ is a invariant differential of a global minimal Weierstrass model.
A better way of formulating the conjecture especially if $E$ is no longer defined over $\mathbb{Q}$ but over an arbitrary global field was given Tate. (See for instance conjecture 2.1 in  Dokchitser's paper for a formulation). Since there are no global minimal models anymore one has to make either a conjecture that is invariant of the choice of a model or work with the Néron model.
