The ring of quaternions is a four dimensional division algebra over the real numbers. They are usually denoted as $\Bbb H$ in honor of the discoverer, William Rowan Hamilton.
The construction of the quaternions was given by Hamilton as follows: take three symbols $i,j,k$ and define $i^2=j^2=k^2=ijk=-1$. As a result, $ij=k$, and $jk=i$ and $ki=j$. Furthermore, $ji=-k$ and $kj=-i$ and $ik=-j$. A quaternion is a linear combination $q=\alpha+\beta i+\gamma j +\delta k$ where $\alpha, \beta,\gamma,\delta\in \Bbb R$. Multiplication between quaternions is carried out by using the distributive rule and the rules for $i,j$ and $k$.
The quaternions turn out to be a noncommutative division ring. In fact, $\Bbb R$ and $\Bbb C$ and $\Bbb H$ are the only associative finite dimensional division rings over $\Bbb R$. They are also the only normed division algebras over $\Bbb R$.