TITLE: What is, exactly, a discrete group?
QUESTION [8 upvotes]: 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 $\mathbb C$.

REPLY [3 votes]: 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.