What is, exactly, a discrete group? What, exactly, is a discrete group? 
In my understanding, a discrete group is a group $G$ on which the only topology 
that can be given is the discrete topology. For example, the group $S^1$ is not discrete because we can give it the topology inherited from $\mathbb C$.
 A: "A discrete group is a group equipped with the discrete topology." http://en.wikipedia.org/wiki/Discrete_group
If a set has more than one element then it can be given a non-discrete topology and so it does not make sense to require that "the only topology that can be given is the discrete topology".
A: If $(G, \tau)$ is a topological group. Then, G is a discrete topological group if $\tau$ is the discrete topology on G.
A: The term 'discrete' seems to be applied to a topology here.  The unit circle with the euclidean topology is a different topological group from the unit circle with the discretee topology.   
A: In my experience the term discrete group can mean different things in different contexts, and in some contexts it can be slightly ambiguous.
(1) If $G$ is a subgroup of a topological group, then being discrete means that $G$ with its subspace topology is discrete.
(2) If not in case (1), but $G$ acts on a metric or topological space $X$, then being discrete means that all the orbits in $X$ are discrete. If some algebraic property of $G$ is being deduced from the action then you probably want the action to be faithful as well.
(3) If no topology or action is mentioned, then discrete likely means finitely generated.
A: In the setting in which the phrase would be used, $G$ is not simply a group, but a topological group.  A discrete group is a topological group in which the topology is discrete.
For example, let us look at the reals under addition, but equip the reals with the discrete topology.  This gives us a topological group, which by definition is discrete.
The fact that the reals can be equipped with a non-discrete topology (such as the usual one) which is compatible with addition is not relevant.
A: The point here is as I see it that in most of cases, the term is used rather for subgroups. We call a subgroup $H$ of a topological group $G$ discrete if the induced topology on $H$ is the discrete one. Also, when one refers to a discrete group it very often means that that the group in question is embeddedable naturally in some topological group (whose topology is quite indiscrete) wherein the former group is a discrete subgroup. 
