What is an example of a force of attraction between two masses?

There are at least three forces that have an attractive aspect to them: the strong force, the electromagnetic force, and gravity. All three of these involve particles with mass, so all three meet the criteria of the question.
The strong force is what holds a nucleus together despite the repulsive positive charges of its protons. At the smaller scale, it also holds the quarks that compose the protons and neutrons themselves.
Electromagnetic attraction only takes place if the masses have opposite electrical charges, although this depends strongly upon the spin state of the masses as well; particles with integer spin can attract like charges instead of repelling them, but the possibility for attraction between masses remains, so this is simply picking at details.
Gravity is probably the most obvious of the attractive forces, and always involves attraction between masses. Our current understanding and theory is that gravity is always attractive, at least in part due to the aforementioned spin state of the gravitational interaction; gravity has a "charge." Because all other known masses share this integer charge, they all attract. 
http://hyperphysics.phy-astr.gsu.edu/hbase/Forces/funfor.html