A sequence over a finite alphabet (one- or two-sided, though I'll show only the one-sided case) x1, x2, ... is called recurrent if every word w=xm...xm+l-1 that appears in it once appears in it infinitely often.
That is, for every N there exists k>N for which w=xk...xk+l-1.
Every periodic sequence is recurrent. Every recurrent sequence is strongly recurrent (but the converse is of course false). See that node for examples of (strongly) recurrent sequences that are not periodic.
To generalize, note that a finite alphabet has only a single metric topology; two sequences are close if they share a long common initial prefix.
So we can use a more interesting space by adding in a metric. A sequence x1, x2, ... over any metric space X (again, one- or two-sided, but I'll only show the one-sided case) is called recurrent if it approaches itself infinitely often, that is if for every n and ε,
d((xn, xn+1, ...), (xn+k, xn+k+1, ...)) ≤ ε
for infinitely many values k.
In other words, if the trajectory of a point x visits any point σnx, then it visits any neighborhood of x infinitely often.