Short answer: lack of a navy, and fear of US intervention.

Long answer:

The Chinese Communist Party (CCP) under Mao desired and planned to invade Taiwan from the inception of the People's Republic of China in 1949 and only gradually abandoned the idea after it became clear it would be almost impossible to do so due to the shifting geopolitics of the Cold War in the 1950s.

The CCP's desire to reclaim Taiwan had nothing to do with Chiang's brutality. Rather, the CCP under Mao had become a fundamentally nationalistic organisation that rose to power on the promise of restoring Chinese nationalism. It saw its historic mission as restoring Chinese sovereignty, and the easiest way to imagine this was returning China to its pre-1839 borders before the First Opium War.

At the time that the People's Republic of China was declared in October 1949, the CCP was still far from achieving its goal of reuniting China: it controlled Manchuria, the North China Plain, and most of the coastal provinces. However, warlords and Kuomintang remnants controlled the southern provinces, Mongolia, Tibet and Xinjiang had declared independence in 1911, and the Kuomintang government had evacuated - with the entire air force, navy and the most disciplined remaining military forces - to the island of Taiwan.

It's true that the Communists had numbers on their side - the PLA numbered something like 5 million men in 1949 - but it was not a mechanised army supported by a powerful industrial base. The PLA's tanks, artillery and aircraft were either ex-Japanese hardware handed over by departing Soviet occupation forces, or captured from the Kuomintang's armies. The PLA Air Force in 1949 had precisely 10 aircraft, for instance. Thus, while the PLA could confidently overwhelm demoralised Kuomintang forces in a land battle through sheer force of numbers, they lacked the materiel needed to stage a maritime invasion.

Consequently, the initial priorities of the PLA were to recapture provinces of mainland China - first Guangdong, then Yunnan, then Tibet, then Xinjiang. (No invasion of Mongolia was considered as it was a Soviet client state, and the new PRC hoped to receive aid from the USSR.) These objectives were achieved by 1951.

In mid 1950, the PLA began staging soldiers on the Fujian coastline and investigating the practicalities of a maritime invasion - such as buying up or acquiring sufficient ships to transport an army across the Taiwan Strait. Fatefully, though, the Korean War intervened.

Before the Korean War, the USA had an ambivalent attitude towards China. On the one hand, US President Harry Truman was clearly anti-Communist and he had provided both financial and military aid to the Kuomintang regime during the Chinese Civil War. This reflected the broader Cold War priorities of the Truman Doctrine (to contain Communism) as well as building on the efforts of his predecessor, Frankling Roosevelt, to forge strong US-Chinese relations. On the other hand, Chiang Kai-shek's regime was corrupt and squandered the US aid it was provided, and the war seemed basically unwinnable anyway - Mao had popular support, and Chiang did not; Mao had skilled generals, and Chiang did not; Mao had a positive vision for China's future, while Chiang had 22 years of broken promises as China's ruler. So, the Truman administration made the decision to 'let China fall', although it continued to recognise Chiang Kai-shek as the legitimate ruler of all of China, and accepted Chiang's claim that his government-in-exile on Taiwan would one day invade China and liberate it from Communism.

Truman's Republican opponents in Congress seized on the Kuomintang's defeat and began to witchhunt to try and find who in the Truman administration was responsible for 'losing' China to the Communists. Remember, this was the era of McCarthyism - the careers of American intelligence and diplomatic officials who worked in China, like John S. Service - were subsequently ruined in the McCarthyist witchhunt. So, even before the Korean War broke out, Truman was under pressure domestically to prove his anti-Communist credentials.

When North Korea invaded South Korea in June 1950, Truman thus felt compelled to expand his policy of containing Communism, and to take an energetic stand against the Chinese Communists. He had already made public commitments to South Korea, Japan and the Philippines - now it seemed logical to make a commitment to Taiwan as a bulwark against the spread of Communism into Asia. Thus, even as Mao's PLA was trying to work out the logistics of how to stage a maritime invasion, Truman ordered the US Seventh Fleet to take up stations in the Taiwan Strait, directly between the Kuomintang army on Taiwan and the Communist forces in Fujian. The message was clear: the US may have been willing to let Mao drive Chiang Kai-shek out of China, but they would not let Mao seize Taiwan. US conviction that it must defend Taiwan was reinforced by China's decision to intervene directly in the Korean War in October 1950; China was now a belligerent opposed to the national security interests of the USA and its allies, and thus any state threatened by China was - by corrolary - a prospective US ally to be brought under the shield of the Truman Doctrine and containment.

Truman's successors, who inherited Truman's Cold War, maintained the same policy. While the US never developed a formal mutual defence treaty with the Republic of China (Taiwan), but until 1971, it acted to defend Taiwan diplomatically in the UN Securitt Council and militarily with the Seventh Fleet any time it was threatened by the People's Republic of China (China) or USSR. More or less every time that the PLA began staging its forces in Fujian for military exercises, or it began shelling the Taiwanese-held islands of Quemoy and Matsu, the US Seventh Fleet would turn up and quietly remind the Chinese government that the most massive navy and air force on earth stood ready to defend Taiwan.

This is one of those questions, by the way, where there is a wonderful confluence between the historical and contemporary. Because, of course, it is still the stated policy of the PRC that Taiwan is an integral part of the People's Republic of China which will one day be reunited. In 2005, the National People's Congress passed the Anti-Secession Law, which basically says that the PRC reserves the right to invade Taiwan and integrate it by force if the Taiwanese choose to declare independence from China. The PLA stages annual military exercises where they simulate maritime invasions, and commandos storm a simulated mockup of the Taiwanese Presidential Palace. Xi Jinping has made military modernisation of the PLA a major objective - such that the PLA might conceivably be able to seize the island of Taiwan swiftly enough that it could present it to the world as a fait accompli before US or international intervention could be taken to stop it. But the prospect of war remains remote - and this is because even though today the US nominally recognises Chinese sovereigny over Taiwan, it continues to maintain a policy of strategic ambiguity about whether it would intervene to defend the island, and it continues to authorise arms sales to the island so that it could defend itself from a mainland attack.

Main source: Jonathan Fenby's Penguin History of Modern China: The Fall and Rise of a Great Power.

/u/theshadowdawn covers most of the points, but it is important to note that Mao did in fact attempt an amphibious invasion in an attempt to get at Taiwan. Between the mainland and Taiwan lie several islands off of the coast of Fujian province, most notably the island of Kinmen (Quemoy) which to this date remain under the control of the Republic of China and not the PRC. The PLA attempted an amphibious invasion of these islands in late 1949 in an attempt to secure them so that the path to Taiwan was clear. The invasion was poorly planned out and the entire landing force-having essentially commandeered whatever junks were around-was quickly surrounded and forced to surrender. While the KMT suffered massive defections in its land forces, for the most part aside from the mutiny of the light cruiser Chung King (which was promptly sunk by air attack) the KMT retained most control of what naval and air assets they had. The defeat at Kinmen combined with lack of amphibious experience, proper equipment, and lack of time before US intervention beginning with the Korean War, essentially preserved Chiang's regime for the duration.

Taiwan certainly had several elements of communist sympathizers and a significant portion of the population was alienated by KMT rule and brutality in the wake of the reclamation. However Chiang's regime was extremely efficient-one would say overly so-in rooting out Communist sympathizers, genuine or merely suspected. This paranoia reached even Chiang's own loyal forces whom were routinely investigated for any signs of political vulnerability. Children of KMT officials and soldiers were often asked questions by poltiical officers-"Does your dad play mahjong?" being one such question to see if an official had a gambling issue (thus being susceptible to financial pressure) and the propaganda machine was out in full force.