The definition of a torus Have seen the term torus used almost in  any article group-theory/algebraic geometry. However, with no definition (presumably it is well known entity)
Nevertheless couldn't find explanation or definition.
Will appreciate explanation to this term (torus) 
 A: So there are two different competing definitions of a torus.
The first of these is the more commonly seen one (I would guess): A torus is a thing which is homeomorphic to a product of circles $S^1 \times \cdots \times S^1$. These are equivalently written as $T_k = \mathbb{R}^k / \mathbb{Z}^k$, where we consider any non-degenerate copy of the lattice of integers inside the reals (i.e. so that it doesn't sit inside some smaller $\mathbb{R}^{n < k}$.
The prototypical example, of course, is the 2-torus $S^1 \times S^1$, which is just the surface of a donut. 
These are great objects to study with a lot of rich mathematics behind them, and in particular when we are looking at even dimensional tori we can try to understand what types of complex structures we can put on them (since in that case, we can write them as a $\mathbb{C}^n / \mathbb{Z}^{2n}$). In the simplest case ($k = 2, n = 1$) we have a one-dimensional complex manifold which is an elliptic curve.
The other definition of a torus is (as said in the comments) and algebraic torus. If you choose a field $k$, then an algebraic torus is given by some product $k^\times \times \cdots \times k^\times$ of the group of invertible elements of that field. For example, if $k = \mathbb{C}$, then $\mathbb{C}^\times$ is an algebraic torus. These lead in particular to the study of toric varieties, which are a collection of algebraic varieties that are combinatorially defined and are often much easier to work with.
From a topological standpoint, and over the complex numbers, an algebraic torus is homotopic to a torus in the other sense. But they are definitely different objects.
