Who decides if someone can become a US Citizen?

There is no single person who makes this decision; it's a bureaucratic process. Any person born on US soil is a citizen by default. Any person who is a legal resident alien can make an application to become a naturalized citizen, and if the application is approved and the person fulfills the requirements they are a citizen. People can become citizens if they are married or related to current citizens. In certain cases people are granted citizenship for humanitarian reasons - usually if there's a credible threat to their lives if they return to their native land. There are restrictions placed on who can become a resident alien - quotas that prevent mass migrations from impoverished nations, screenings to weed out dangerous criminals, and the like - but aside from that the process is largely a matter of crossing T's and dotting I's.