What is an sn2 reaction and how does it work on this reaction of a hydroxide ion (OH -) (nucleophilic) reacts with a molecule of bromomethane...

Sn2 is an acronym that stands for Substitution Nucelophilic Bimolecular (the 2 d two molecules).  It describes a reaction mechanism whereby one molecules approaches and bonds with another molecule while simultaneously ejecting another atom or group of atoms from the molecule as a replacement.  Let's look at the example you provide with the reaction of a hydroxide anion (OH-) with bromomethane. 

OH- + BrCH3 ---> ?

In this case, the hydroxide anion is a nucleophile...

Sn2 is an acronym that stands for Substitution Nucelophilic Bimolecular (the 2 d two molecules).  It describes a reaction mechanism whereby one molecules approaches and bonds with another molecule while simultaneously ejecting another atom or group of atoms from the molecule as a replacement.  Let's look at the example you provide with the reaction of a hydroxide anion (OH-) with bromomethane. 

OH- + BrCH3 ---> ?

In this case, the hydroxide anion is a nucleophile (literally a lover of atomic nucleii).  This means that it is a negatively charged species that is looking for a positively charged species to interact with.  The bromomethane is the electrophile (literally lover of electrons).  This means that it is a partially positively charged species that is available to react with a nucleophile.  The partially positive charge in bromomethane results from the electronegativity of the bromine atom.  The bromine atom is a highly electronegative atom, meaning that it will pull electron density toward it.  So the electrons in the covalent bond between the carbon and bromine are pulled somewhat toward the bromine atom, thus putting a partial positive charge on the carbon atom.  So when the hydroxide anion approaches the bromomethane for a chemical reaction, this is called the nucleophilic attack since the negatively charged oxygen anion is "attacking" the partially charged carbon atom for the SN2 reaction.  A new covalent bond formed between the nucleophile and the electrophile occurs at the same time that bromine atom starts to lose its bond with the carbon and becomes the leaving group.  This results in a fleeting species called a transition state where the new bond is partially formed and the old bond is partially broken.  This transition state immediately breaks down to for the products, in this case methanol (CH3OH) and the bromide anion

HO- + CH3Br ------> HO--CH3--Br -------> HOCH3 + Br-

So ultimately the bromine atom on the CH3 group is substituted with the hydroxyl group.  The energy diagram for an SN2 reaction is shown in the attachment below.  The energy level of the starting reagents is on the left.  The transition state is formed in the middle which is a higher energy species.  This then goes to the products on the right which is the lowest energy level.  The overall drop in internal energy from the starting reagents to the products is the driving force for the reaction.