In chemistry, base can be in a way seen as an "opposite" of acid. Acid is any substance that gives out hydrogen ions, base is any substance that takes in them. This is the Brønsted-Lowry theory. Bases have a pH greater than 7. (See pH for an explanation.) There are two groups of bases: the weak bases and the strong bases or alkali.
For example, ammonia is a weak base. An ammonia molecule contains one nitrogen atom bonded to three hydrogen atoms. One pair of electrons in the nitrogen atom is not bonded to hydrogen. Because they're negative in electric charge, the molecule is polar. (See polarity) The pair of electrons attracts positive hydrogen ions from the solution. So, the hydrogen ion bonds to the pair of electrons with a dative bond. The ammonia has taken in hydrogen ions.
Actually this is a simplification. The ammonia molecules is quite stable as is, but some of them take one hydrogen nucleus from water and leave a hydroxide ion behind. The reason why ammonia neutralises acid is that a dative bond to a H+ in an ammonium ion is more stable than it's in an oxonium ion. (Remember, there are no naked protons in an acidic water solution, but H3O+ ions.) When an ammonia molecule hits an oxonium ion, it takes its extra proton and neutralises it to water.
The other group of bases is called alkali or strong bases. They release active hydroxide ions (OH-) to the solution, because they dissociate completely. These ions can react with the hydrogen ions, giving water. Alkalis are stronger bases than the other bases, because the formation of water, the neutralisation, is almost an irreversible reaction, whereas the dative bond to a molecule is relatively easy to reverse. Examples: sodium hydroxide, other alkali hydroxides, earth alkali metal hydroxides.