In chemistry a strong base is, basically, the basic analogue of a strong acid. That doesn't tell you much, does it?
From a chemical standpoint, strong bases are able to yank a proton off weak acids or strong acids rather easily. If they are in water, they undergo complete and utter dissociation. For example, sodium hydroxide
NaOH(aq) --> Na+(aq) + OH-(aq)
No equilibrium, just utter dissociation.
What strong bases are there?
What sort of properties do they have?
Generally? Like most bases, they should have a bitter taste. But I really don't recommend gulping down a bottle of sodium hydroxide anytime soon. Rather generally, they also feel slippery.
Another general rule about them is they have a small pKb (and thus, a large base dissociation constant). This is because they dissociate almost completely in water (numerous factors influence why they dissociate so well).
Very much like strong acids, the conjugate acid of a strong base is PATHETIC! Let's use the example of lithium hydroxide.
LiOH --> Li+ + OH-
That happens almost all the way. Very little LiOH remains. However:
Li+ + H2O --> LiOH + H+
THAT REACTION DOES NOT HAPPEN, CANNOT HAPPEN! Except maybe in almost negligible amounts, but seriously. The lithium ion is so pathetic as an acid (accepting the hydroxide ion) that that reaction barely happens at all. The conjugate acid of a strong base just plain doesn't hydrolyze in water.
How they react with stuff
The strong bases generally react with strong acids to yield a salt and something else. In the case of metal hydroxide strong bases, this "something else" is usually water.
Strong bases are also capable of very easily deprotonating (yanking off a proton from) other molecules. They neutralize other acids, even weak acids. They can also take away protons from other substances, depending on the substance and the strong base used. Some strong bases are stronger than other. For example, NaH is a "very strong" base, but it reacts explosively with water.
For metal hydroxide strong bases, the reaction in water is, generally, a dissociation one:
SOH --> S+ + OH-
Where S+ is the metal. This also happens for calcium hydroxide and barium hydroxide, only they go completely to Ba+2, Ca+2, and 2 OH-.
For sodium hydride, the reaction with water basically takes a hydrogen from water, which acts as an acid:
NaH + H2O --> Na+ + H2(g) + OH-
As mentioned earlier, this reaction is explosive.
Often, strong bases that have been standardized (the concentration of the base is known) are used in titrations with solutions of unknown concentration, to determine their concentration.
My AP Chemistry teacher
http://en.wikipedia.org/wiki/Strong_base (Revision from December 6, 2005)