The four basic reaction types can be outlined as follows:
A + B -> AB
AB -> A + B
Note that the synthesis and decomposition reactions are exact opposites of one another; if you run either of them in reverse, you get the other type of reaction.
Single Displacement Reaction:
AB + C -> AC + B
Double Displacement Reaction (also known as a Metathetical Reaction):
AB + CD -> AC + BD
The reaction you are asking about is a single displacement reaction because it follows the single displacement pattern; in your example the H2O breaks up, and some (though not all) of the H is replaced by Na, forming the new compound NaOH. The remaining H atoms join to form H2 gas. Since this shows an atom being replaced by another atom and not forming a new compound, it qualifies as a single displacement. Chemists generally consider diatomic gases such as H2, O2, Cl2, and so on as similar to single atoms in this case.
There exist six types of chemical reactions, one of which is a Single Displacement reaction. That occurs when one reactant element trades places with another of the reactant elements in a compound.
The general formula is:
A + BC --> AC + B
In your example, two sodium atoms react with water to make two sodium hydroxide molecules and hydrogen:
2Na + 2H2O --> 2NaOH +H2
which follows the form exactly, so the reaction is a single displacement. See the link for a complete description of the other 5 types of reactions: