In general, a group that is isomorphic to a $d$-dimensional torus over the algebraic closure of the ground field can be identified by the action of the Galois group on its character group / group of one-parameter subgroups, which is $\mathbb Z^d$, so you get a homomorphism $Gal(\bar{k}/k) \to GL_d(\mathbb Z)$. In the one-dimensional case, this is just a quadratic character, so tori correspond to quadratic fields, in the manner discussed in the comments. In particular, all $1$-dimensional tori in any group embed into $GL_2$.
The reason you can recover it from the Galois action on the character group is that the points of the torus are exactly the homomorphisms from the character group to $GL_1(\bar{k})$ that are Galois-equivariant. This is a version of Pontryagin duality.
This is part of the general theory that twists of an object $X$ over a field $k$ are classified by the Galois cohomology group $H^1(k,X)$, or, for twists over an arbitrary base, an etale cohomology group.
EDIT: For a very explicit description, the Galois rep associated to the quadratic field extension $\mathbb Q(\sqrt{D})$ is the group of determinant $1$ matrices that preserve the quadratic form $x^2-Dy^2$, which means
$\left(\begin{array}{cc} a & Db \\ b & a \end{array}\right) $
for $a^2-D b^2 =1 $
for $D=-1$ this reduces to the classic $SO_2$, and for $D=1$ this is just a regular split torus.