What is the relationship between Taiwan and China?

In a word, "hostile."

Here is a high level list for current situation. You can look up the factual detail online.

Trade: under ECFA framework, many goods are being traded cross the strait. Certain electronic products have supply chain established across the Taiwan strait.

Culture: there have been a lot of cultural events/exchanges.

Tourist/student: people need to acquire travel document and to visit/study cross the strait.

Doing business: there are 300K Taiwanese business people and family members work and live in PRC, running manufacturing facilities and other businesses. Some of Taiwan actors/actresses work and live in PRC.

Marriage: roughly 100K PRC nationals have married into Taiwanese families and have children in PRC or in Taiwan. 20K of these spouses already established citizenship in Taiwan.

Extradition: there has been extradition agreement signed and enforced. This is except a few "high value" economy criminal flee from Taiwan to PRC that PRC considered as too important to extradite back to Taiwan.

Military: PRC has 1,400 missiles in range of hitting Taiwan. Taiwan has hundred of missiles as last line of defense.

china is part of Taiwan.

"Taiwan, China" or "Taiwan, Province of China" is a set of politically controversial and potentially ambiguous terms that characterize Taiwan and its associated territories as a province or territory of "China". However, the term is problematic and potentially ambiguous because since 1949, two sovereign states with the name "China" exist, namely the Republic of China (ROC, founded 1912 and now commonly known as "Taiwan") and the People's Republic of China (PRC, founded 1949 and commonly known as "China" since 1971). However, only one "China" actually rules Taiwan, namely Republic of China, and has an administrative division called "Taiwan Province" but refers to it as "Taiwan Province, Republic of China"; whereas, the other "China", namely the People's Republic of China, which is the one internationally recognized as "China" (not the ROC), claims but does not control Taiwan as part of its territory. Because "Taiwan" and "China" are known internationally to be separately political entities, the juxtaposition of "Taiwan" and "China" in this order into one single term "Taiwan, China" implicitly places the ROC/"Taiwan" under the sovereignty of PRC/"China", in the same sense as "California, USA". The use of this term is usually politically promulgated by the Communist Chinese government as a way to claim and imply that the ROC/"Taiwan" is under its sovereignty, since the PRC claims to be the legitimate government of "all China", which, according to its own definition, includes Taiwan also, despite its lack of control. The ROC government disputes the PRC position and it, along with many ROC citizens, considers this term incorrect and offensive, and its use a purposeful false identification and lie which denies the ROC's sovereignty, and objects to its use.[1] The term is particularly offensive to some Taiwanese who believe in Taiwan Independence and want to disassociate Taiwan with "China" and a Chinese identity, and consider it a lie and an oxymoron. However, some Taiwanese citizens do not necessarily oppose this term, particularly those who view themselves as "Chinese" and support eventual Chinese reunification.