Let me add a few remarks to Qiaochu's very nice answer:

• The parity conjecture (i.e. algebraic rank equals analytic rank modulo 2) is known for all elliptic curves over number fields under the assumption that Tate-Shafarevich groups of elliptic curves over number fields are finite. The survey that Qiaochu has linked to describes the proof.

• There are refined parity conjectures for twists by Artin representations. I will also take this opportunity to fill Qiaochu's (ii) with some more details. Let $A/K$ be an Abelian variety and let $\tau$ be an Artin representation of $G_K$. Let $p$ be a prime number. Consider the Pontryagin dual of the $p^{\infty}$-Selmer group of $A/K$ and take the tensor product with $\mathbb{Q}_p$, call this $\chi_p(A/K)$. This is a $\mathbb{Q}_p$-vector space. If we believe that the $p$-primary part of the Tate-Shafarevich group of $A/K$ is finite, then the $\mathbb{Q}_p$-dimension of $\chi_p(A/K)$ is exactly the rank of $A(K)$. If we don't assume this, then we have to allow for the possibility of some copies of $\mathbb{Q}_p/\mathbb{Z}_p$ inside the Tate-Shafarevich group increasing the dimension. In any case, $\chi_p(A/K)$ is a $G_K$-representation, and we can consider the number of copies of $\tau$ inside it: $\langle\tau,\chi_p(A/K)\rangle$. On the analytic side, we have the twisted $L$-function $L(A/K,\tau,s)$ and its root number $w(A/K,\tau)$. The $p$-parity conjecture for twists now predicts that $$ (-1)^{\langle\tau,\chi_p(A/K)\rangle} = w(A/K,\tau). $$ If we believe in the finiteness of Tate-Shafarevich groups, we could instead work with the $\tau$-isotypical component of the Mordell-Weil group $A(K)$. Anyway, what I wanted to say is that we now know the $p$-parity conjecture for various different twists, here are some examples: DD1, Theorems 1.3, 1.11, 1.12, DD2, Theorem 1.11.

Let me add a few remarks to the very nice CW answer already given.

• The parity conjecture (i.e. algebraic rank equals analytic rank modulo 2) is known for all elliptic curves over number fields (not just over $\mathbb{Q}$) under the assumption that Tate-Shafarevich groups of elliptic curves over number fields are finite. The survey by Tim Dokchitser that has already been linked to describes the proof.

• There are refined parity conjectures for twists by Artin representations. I will also take this opportunity to explain the content of the $p$-parity conjectures in a little more detail. Let $A/K$ be an Abelian variety and let $\tau$ be an Artin representation of $G_K$. Let $p$ be a prime number. Consider the Pontryagin dual of the $p^{\infty}$-Selmer group of $A/K$ and take the tensor product with $\mathbb{Q}_p$, call this $\chi_p(A/K)$. This is a $\mathbb{Q}_p$-vector space. If we believe that the $p$-primary part of the Tate-Shafarevich group of $A/K$ is finite, then the $\mathbb{Q}_p$-dimension of $\chi_p(A/K)$ is exactly the rank of $A(K)$. If we don't assume this, then we have to allow for the possibility of some copies of $\mathbb{Q}_p/\mathbb{Z}_p$ inside the Tate-Shafarevich group increasing the dimension. In any case, $\chi_p(A/K)$ is a $G_K$-representation, and we can consider the number of copies of $\tau$ inside it: $\langle\tau,\chi_p(A/K)\rangle$. On the analytic side, we have the twisted $L$-function $L(A/K,\tau,s)$ and its root number $w(A/K,\tau)$. The $p$-parity conjecture for twists now predicts that $$ (-1)^{\langle\tau,\chi_p(A/K)\rangle} = w(A/K,\tau). $$ If we believe in the finiteness of Tate-Shafarevich groups, we could instead work with the $\tau$-isotypical component of the Mordell-Weil group $A(K)$. Anyway, what I wanted to say is that we now know the $p$-parity conjecture for various different twists, here are some examples: DD1, Theorems 1.3, 1.11, 1.12, DD2, Theorem 1.11.