What does it mean for rational numbers to be "dense in the reals?" What does it mean for rational numbers to be "dense in the reals?" I can't seem to find a decent explanation online...
It means that between any two reals there is a rational number.  The integers, for example, are not dense in the reals because one can find two reals with no integers between them.

That definition works well when the set is linearly ordered, but one may also say that the set of rational points, i.e. points with rational coordinates, in the plane is dense in the plane.  Then it must be defined differently: it means that every open set in the plane intersects the set of all rational points.  No matter how small you make an open disk in the plane, it cannot avoid containing some rational points; so the set of all rational points is dense in the plane.  In the case of the line, saying that every open interval contains some rationals amounts to the same thing as saying that between any two reals there is some rational number between them.
Between any two rational numbers there exist another rational number. For example 1/2 and 1/4 are two rational numbers, but there exist another rational number 1/3 between the two above.In the case of other subsets of numbers in real numbers for instance,integers,there cannot exist another integers between any two.