The motivation for the embargo was the expropriation of US property within Cuba without payment. And payment has not been tendered in the 70 years since, so technically the underlying issue still remains.
In practice, the issue is really "the communist government of Cuba still exists."
In June 1960 a key incident occurred: Eisenhower's government refused to export oil to the island, leaving Cuba reliant on Soviet crude oil. Cuba and the Soviet Union signed a trade agreement according to which the Soviet Union would provide 900,000 tons of oil to Cuba. The United States viewed the agreement as a provocation, and successfully urged Esso, Texaco, and Shell to refuse to process Soviet crude in their Havana and Santiago de Cuba refineries. On June 29 and July 1, 1960, Cuba confiscated the refineries.
[Edit] Ok, I see the parent comment is an uncited quote of the Wikipedia page "United_States_embargo_against_Cuba." Like I said below, Wikipedia is oddly lacking in a citation for this action by "Eisenhower's government."
> Eisenhower's government refused to export oil to the island
I did a hunt for a cite to support this claim, but did not find one. I searched national archives for executive orders [0, 1] and didn't find anything. The Wikipedia pages that have statements like "In June 1960 a key incident occurred: Eisenhower's government refused to export oil to the island" [2] and "This nationalization of property resulted partially from U.S. President Dwight D. Eisenhower's decision to cancel all oil exports to Cuba." [3] are curiously lacking in a citation.
I wish I still had my copy of The Prize which I recall had excellent footnotes and references.
In practice, the issue is really "the communist government of Cuba still exists."