The ordering from best nucleophile to worst nucleophile is as follows:

Smaller molecules are better nucleophiles than larger ones (they are not as sterically hindered).  is a better nucleophile than because nitrogen is less electronegative than oxygen (Look for the the lower electronegativity on the atom holding the lone pair of electrons). Thus, the lone pair of electrons on nitrogen are not as stable as those on oxygen, and will readily donate its pair of electrons to another species. 

The larger the atom, not molecule, the better the better the nucleophile . Atomic radius increases as you go down a group on the periodic table. Nucleophilicity can also be determined according to strength of the anion as a conjugate base. Remember that  is the strongest acid and thus has the weakest/least stable conjugate base.

 is least nucleophilic because its oxygen has no negative charge, and oxygen is more electronegative than sulfur, thus the lone pair of electrons on oxygen are more stable than they would be on sulfur or nitrogen.

 forms strong hydrogen bonds with polar protic solvents, which makes it less able to interact with any target molecules (electrophiles). Note: in a polar aprotic solvent, hydrogen bonds cannot form, and  is more nucleophilic than ^.

A Lewis base is a compound that donates a pair of electrons. The only compound shown that possesses an electron pair that can be donated is ammonia.  is the Lewis base.  and  are Lewis acids and are therefore wrong. Neither carbon dioxide nor methane are Lewis bases.

A good nucleophile should not be bulky. If the nucleophile is bulky, the compound will not be able to reach the carbonyl carbon, where the reaction occurs. With increased steric hindrance due to a bulkier nucleophile, the reaction will run slowly, and the compound is not a good nucleophile. Good nucleophiles also want to give away electrons so that the reaction can occur. This means that conjugate bases make better nucleophiles than acids.

To promote elimination rather than substitution, a molecule must be a strong base and weakly nucleophilic. Typically weak nucleophiles are large and bulky, which therefore have difficulty undergoing backside attack. Only tert-butoxide is a weak nucleophile, due to its branching, meaning it is least likely to participate in substitution. 

 is the weakest base as its negative charge is spread out more evenly over the large atom. Thus, it is the best leaving group of these four halogens. It is the best leaving group due to its largest size and distribution of negative charge. 

Note that many of these carbonyl groups are actually part of various functional groups. For example, the gold is an aldehyde, the green and purple are both ketones, the red is an amide, and the blue is an ester. We know the electrophilicity of carbonyl-containing functional groups is as follows:

Thus, our aldehyde, in gold, is the most electrophilic.

A Lewis acid is an electron pair acceptor. A Lewis base is an electron pair donor.

 is an ionic molecule and is neutral.  and  have positive charges and are electron pair acceptors, as they are more stable when they are neutral.

The correct answer, , has a lone pair on the nitrogen atom that can be donated to form bonds with other atoms, so it is a Lewis base.