The circle group $\mathbb{T}$ is equivalently (isomorphically)
the quotient group $\mathbb{R}/\mathbb{Z}$ of the additive group of real numbers by the additive group of integers, induced by the canonical embedding $\mathbb{Z} \hookrightarrow \mathbb{R}$;
the unitary group $\mathrm{U}(1)$;
the special orthogonal group $SO(2)$;
the subgroup of the group $\mathbb{C}^\times$ of units of the field of complex numbers given by those of any fixed positive modulus (standardly $1$).
For general abstract properties usually the first characterization is the most important one. Notably it implies that the circle group fits into a short exact sequence
(On the other hand, the last characterization is usually preferred when one wants to be concrete.)
A character of an abelian group $A$ is simply a homomorphism from $A$ to the circle group.
A principal bundle with structure group the circle group is a circle bundle. The canonically corresponding associated bundle under the standard representation of $U(1) \hookrightarrow \mathbb{C}$ is a complex line bundle.