From my knowledge, as you go down the periodic table, the elements become more and more reactive because there are more particles bunched up together in the nucleus and this causes the atom to be very reactive. Since co-60 is someway down the table, it is fairly reactive.
Cobalt-60 is unstable because there is a high neutron to proton ratio. It has 33 neutrons and only 27 protons. It is formed by bombarding Cobalt-59 with a neutron.
This web site is very good at explaining Cobalt-60:
Cobalt-60 is a radioactive form of normal cobalt, and hence isn't technically in the periodic table.
The reactivity you are referring to is ionisation energy. Ionisation energy increases from left to right of the periodic table and increases up the periodic table. This is because atoms towards the upper right of the periodic table are smaller and therefore harder to remove the electrons from the atom (hence, they are less reactive). Atoms to the lower left are much larger and cannot hold their electrons as strongly; consequently, their electrons are removed much easily meaning they are much more reactive. Normal cobalt is located towards the middle of the periodic table; it isn't really reactive.