We cannot go all the way "up to now" due to the 20 year rule, and there have been major shifts in popular opinion and some positions, though the fundamentals have not changed since 2002. This is a HUGE topic with lots of things I'd love to spend detail on, so I'll try to include a summary at the end, with a more detailed history before.
We'll start by rewinding a few hundred years for some background. Taiwan has been a part of what is considered China proper since at least the Qing Dynasty. A Ming loyalist named Zheng Chenggong (郑成功, also known as Koxinga) had established control over the island in 1661, invading with a force of over 25,000 troops from the Chinese mainland. The family of Zheng would rule the island until 1683, when admiral Shi Lang (施琅) conquered the island on behalf of the Kangxi Emperor.
Taiwan remained under Qing Dynasty control until it was surrendered to Japan in the Treaty of Shimonoseki after the first Sino-Japanese War in 1895 (along with the Liaodong peninsula and Penghu islands; the Senkaku/Diaoyu islands were annexed separately and not named in the treaty, which contributes to their disputed status today).
Taiwan was occupied by Japan from 1895-1945 and called the Republic of Formosa (using the old Portuguese name for the island). During this period, the Qing Dynasty government on the mainland was overthrown by the Xinhai Rebellion and was replaced by the Republic of China in 1911, which inherited the Qing's treaties and territorial claims. The treaties with Japan were abrogated following the Japanese invasion of China and the second Sino-Japanese War (1937-1945), and the Cairo Declaration in 1943 stated the will of the allies to return Taiwan and other lost territories to the Republic of China:
The Three Great Allies are fighting this war to restrain and punish the aggression of Japan. They covet no gain for themselves and have no thought of territorial expansion. It is their purpose that Japan shall be stripped of all the islands in the Pacific which she has seized or occupied since the beginning of the first World War in 1914, and that all the territories Japan has stolen from the Chinese, such as Manchuria, Formosa, and The Pescadores, shall be restored to the Republic of China.
Formosa is of course Taiwan, and Taiwan and the other territories taken by Japan were returned to the Republic of China in October 1945 and Japanese claims were formally renounced in the Treaty of San Francisco (1951) and Treaty of Taipei (1952), though Taiwan was already part of the Republic of China and governed by the same.
Now events on the mainland start to matter- after the surrender of Japan in 1945, a civil war broke out on the Chinese mainland between the Republic of China and the Communist Party. The Communist Party was able to take all of mainland China from the Republic of China by 1949, with Chiang Kai-shek fleeing with his government to Taiwan.
This is when the split occurs- the founding of the People's Republic of China was declared in 1949, while the Republic of China they claimed to replace still survived on Taiwan. The People's Republic claimed to be the successor state to the Republic of China, inheriting all its territorial claims, just as the Republic of China succeeded the Qing Dynasty. Meanwhile, Chiang Kai-shek still ruled in Taiwan, which considered itself the only true government of China, and maintained itself as such, while believing it could retake the mainland. Likewise, the PRC wanted to retake Taiwan, but didn't really have a navy- after the Korean War, and the US deployment of the 7th fleet to the strait, this was simply not possible.
This left the world with two governments both claiming to be the legitimate government of all of China- the Communist PRC on the mainland and the military dictatorship of Chiang Kai-shek on Taiwan.
The next few decades saw growing recognition of China as the legitimate government of China, culminating in the UN Resolution on Admitting Peking, or General Resolution 2758. This saw the PRC take the place of the RoC at the UN in October 1971, with the US voting against. Nixon later visited China in 1972, with the US switching its formal recognition to the PRC in 1979. Over time, more and more countries switched diplomatic recognition to the PRC as the sole legal and legitimate government of "China".
So where did all this leave Taiwan? Simply put, it hadn't officially stopped claiming to be the legitimate government of China and considering the Communist government as a bunch of rebels, but these claims were no longer taken seriously (and internally, it had given up on taking back the mainland by military force). Likewise, the conditions China established for diplomatic relations, including trade, with all other countries including their acknowledgment of Taiwan as a part of China. The "Three Joint Communiqués" that led to the establishment of US-China relations, including the Shanghai Communiqué, stated that the US acknowledged that all Chinese on both sides of the Taiwan Strait maintain there is but one China". This meant that effectively, Taiwan was regarded (officially) as a part of China (though which China, PRC or RoC, was not explicit).
For most of this period, both Taiwan (the Republic of China) and the People's Republic of China agree there is one just one China, but who the real government is simply an internal dispute, the result of a Civil War that never ended. This was formalized with the 1992 Consensus (92共识/九二共識), as both parties stated they both believe there is only one China, but each has their own interpretation of what that means.
However, much of this rested on Taiwan being ruled by the Nationalist Kuomintang (國民黨), the ruling party of the original Republic of China and the military dictatorship of Taiwan. The above 92 Consensus was a product of Lee Teng-hui's government, which also created a National Reunification Council whose mission was to plan to reintegrate the mainland into Taiwan. After Taiwan became a democracy in the 1980s and 1990s, the Democratic Progressive Party (DPP, or 民進黨) became more of a force, finally winning the presidency in 2000 with the election of Chen Shui-bian (we're getting really close to the 20-year rule here).
Here we should take a detour into Taiwanese politics- the Nationalists (Kuomintang, KMT, aka the RoC) were very much outsiders when they fled to Taiwan in 1949. There was a famously brutal massacre of local Taiwanese (the 228 incident), a 30 year reign of terror known as the White Terror, and they were from the mainland. Most local Taiwanese were the descendants of Fujianese who had come to the island, and they did not have much of a voice in politics the same strong, recent connections to the mainland that the Kuomintang did. They had lived as a Japanese colony for 50 years, and had a different political identity than the members of the KMT and mainland Chinese elite that fled to the island in 1949, displacing the local elite. The DPP had a much stronger Taiwanese identity, and though they placed lots of pressure on the ruling KMT in the 1990s, they did not take power until their minority government from 2000-2008, which I can't talk about too much without violating the 20 year rule. Let's just say that the many Taiwanese who identified less strongly with the mainland and reunification did not hold much political power prior to the year 2000, and that the same nationalist government who founded the Republic of China in 1912 and considered itself the rightful government of all of China was calling the shots until then. This is not to say no shifts occurred within the KMT; beyond giving up on retaking the mainland, the RoC adjusted its constitution in 1991 to recognize it no longer controlled the mainland.
Whew. That's a lot of background. Let's summarize:
TL;DR The Republic of China had a civil war from 1945-1949. The Communists defeated the Nationalist government led by Chiang Kai-shek in 1949, who fled to Taiwan (then a province of the Republic of China), who considered himself the legitimate leader of China waiting to take back the mainland. The Communists established the People's Republic of China, and regarded themselves as the new China, and the inheritors of the political legacy and territory of the old China- which included Taiwan, and consider it essential that they complete the "reunification" of China by unifying with Taiwan. However, they couldn't take Taiwan, nor could Taiwan retake the mainland. This left the world with two Chinas who competed for international legitimacy. The People's Republic obtained enough political support to be recognized as the only China, and made recognition of itself as the only China and Taiwan as a part of it a condition of diplomatic relations and trade, leading to other states breaking off relations with Taiwan. Taiwan, still ruled by a party which did not support independence, never officially sought independence, though independence movements gained strength over time and it began to move away from considering itself the official, legitimate government of China.
You could say Taiwan is a country: the Republic of China, with its own foreign policy and military, recognized by a handful of small countries only. But Taiwan is not, as that would break historical continuity and require a declaration of independence and constitutional changes, and is guaranteed to trigger war, as it would be seen by the mainland as a renegade province trying to secede, and the "unfinished" Civil War would then have to be finished. As long as those official positions on "One China" do not change, both parties can dispute who is the real China and it remains an "internal" dispute. Even countries which recognize the Republic of China (Taiwan) do not recognize two Chinas, as it has not declared independence from the People's Republic.