The 1961 Bay of Pigs Invasion relates to the Cold War through the fact that it is one instance that helped escalate the Cold War. The Bay of Pigs invasion was plotted by President Dwight D. Eisenhower in response to Communist Fidel Castro's seizure of power over Cuba in 1959. Eisenhower took the U.S. presidency in 1953 and witnessed Castro seize power, attack U.S. companies in Cuba, and vehemently speak out against capitalist America. Seeing both Communism and Castro as a threat to the U.S., in 1960, just before the end of his second term, Eisenhower commanded the CIA to begin arming and training Cuban exiles in order to launch an attack on Cuba in the hopes of bringing down Castro. The plan had to undergo several revisions, and overseeing the attack was actually passed down to the new President-elect, John F. Kennedy, at the end of Eisenhower's term. Regardless, Eisenhower did review the final plans and did approve them in the same way that Kennedy had approved them. However, the attack went horribly wrong.
One reason why the attack failed is because Cuba was already expecting the invasion. Che Guevara, Castro's commander of Cubans western military forces, had already anticipated the invasion, and decided to ensure that all of Cuba's people were armed for guerrilla warfare. What's more, it later became known that even the CIA knew that the Soviet Union was well aware of the upcoming invasion and was prepared to defend Castro. Even more importantly, most of the Cuban people actually supported Castro, and the success of the invasion rested mostly on being able to incite the Cuban people into a full-scale government insurrection aimed at Castro. As a result, the invasion was met with Castro's military counterattacks, and the U.S. did not have the air support it needed to defend the 1200 men initiating the invasion, resulting in the death of 100 men and the capture of the rest.
More importantly, the failed Bay of Pigs attack had dire lasting consequences. For one thing Castro asked the Soviet Union for more military protection, and the Soviet Union readily complied to protect their communist ally, leading to the Cuban Missile Crisis of 1962 in which the Soviet Union planted nuclear missiles in Cuba aimed directly at the U.S. Naturally, had the U.S. been unsuccessful at negotiating the disarmament of the missiles, the Soviet Union could have turned the Cold War into a full-blown nuclear war, showing us just how much the failed Bay of Pigs invasion escalated the Cold War.
The Bay of Pigs invasion was an attempt by the United States to covertly overthrow the Castro regime. It was an embarrassing calamity for the United States, and a major stain on the legacy of President Kennedy (though President Eisenhower worked with the CIA to develop the plan prior to JFK's inauguration). At the time of the invasion, the United States was not only extremely concerned about the spread of communism but also in protecting the mainland from Soviet attack. Intelligence led U.S. presidents and top diplomats to believe that Cuba had allied with the USSR, and that allowing Castro to stay in power would lead to an attack (a fear proven to be valid during the Cuban Missile Crisis in 1962, though diplomacy prevailed). The Bay of Pigs has a relationship to the Cold War because it displays the major themes of the Cold War--namely, the ideological battle between democracy and communism, the quest to obtain the support of other countries, and the extreme paranoia that existed at the time. The key in understanding the relationship between the Bay of Pigs and the Cold War lies in recognizing the alliance between Cuba and the Soviet Union, with both countries vying to secure ideological and diplomatic power in Cuba. Through the Bay of Pigs invasion, it is possible to see the major themes and conflict present in the Cold War.