$R_y(\theta)$ gate rotates a qubit by angle $\theta$ about the $y$-axis of the Bloch sphere. It is one of the three rotation gates and is defined as the matrix exponential of the Pauli-Y operator:
$$R_y(\theta) = e^{-i\theta Y/2} = \begin{pmatrix}\cos\dfrac{\theta}{2} & -\sin\dfrac{\theta}{2}\\[6pt] \sin\dfrac{\theta}{2} & \cos\dfrac{\theta}{2}\end{pmatrix}$$
Applied to the computational basis states:
$$R_y(\theta)\lvert 0\rangle = \cos\tfrac{\theta}{2}\lvert 0\rangle + \sin\tfrac{\theta}{2}\lvert 1\rangle \qquad R_y(\theta)\lvert 1\rangle = -\sin\tfrac{\theta}{2}\lvert 0\rangle + \cos\tfrac{\theta}{2}\lvert 1\rangle$$
Unlike $R_x$, the matrix entries are entirely real — no imaginary phases appear in the off-diagonal. This makes $R_y$ the natural gate for preparing states with real amplitudes. At $\theta = \pi/2$ it produces $\tfrac{1}{\sqrt{2}}(\lvert 0\rangle + \lvert 1\rangle) = \lvert +\rangle$, the same result as the Hadamard gate up to global phase. At $\theta = \pi$, $R_y(\pi) = -iY$ — a combined bit and phase flip up to global phase. The eigenstates are $\lvert +i\rangle$ and $\lvert -i\rangle$ (the $y$-axis poles of the Bloch sphere).