Which molecules will undergo aromatic bromination (#"Br"_2, "FeBr"_3#) the fastest? Why?

The #-"OH"# group causes the aromatic ring to be strongly activated. The reaction can undergo at a reasonable rate without the need of #"Fe"# nor #"FeBr"_3# catalyst.

Aromatic bromination is an electrophilic aromatic substitution (EAS) reaction, which will require benzene to act as a nucleophile to acquire an electrophile.

Therefore, any directing groups that activate the ring will make it react more quickly with respect to aromatic bromination.

Activating groups are more electron-donating groups (EDGs), and deactivating groups are more electron-withdrawing groups (EWGs).

• Phenol, having a good EDG, strongly activates the ring.
• Anisole, having an ether substituent, has hyperconjugation between the oxygen's filled #pi# orbital electrons and the methyl's #"C"-"H"# #sigma#-bonding electron pair as a opposing effect to the electron donating behavior of #-"OR"#, which weakens the electron-donating capacity of the substituent, giving anisole a less strongly-activated ring.
• Diphenyl ether, having only one ether group sharing two aromatic rings, only activates one of the rings. Not a great candidate, and worse than anisole.
• Acetanilide, having a decent EWG, is a moderately-activated ring.
• 4-bromophenol has both an #-"OH"# and a #-"Br"# on opposite sides. #-"OH"# is a strongly-activating group (high-lying #sigma# MO), but #-"Br"# is a weakly-deactivating group (due to its filled #pi^"*"#, rather than filled #pi# MOs). These, being para to each other, counteract each other and make the ring overall approximately moderately-activated, depending on the extent of deactivation. Either way, not the best.

Therefore, since phenol contains the most strongly-activated ring towards EAS, it participates fastest in EAS.