Zheng He served as fleet admiral under the Yongle Emperor. He conducted seven ‘Treasure Voyages’ with the Ming Treasure fleet, as a massive tour de force. These voyages went to various places in South East Asia, as well as India, and as far as East Africa and the Sinai peninsula (Map). The aim of these voyages was to gain tribute from the various kingdoms considered under the ‘sphere’ of China; and thus the size of the fleet (supposedly one of the largest navies ever built with roughly 100-240 ships of various size) was impressive. The largest ships, the Treasure ships, were very large indeed, and some have argued that the largest ships in the fleet were over 100 metres long; however this is largely unverifiable, and is very questionable as it would make them at least twice the length of European vessels of the same period (thus it’s questionable whether construction of such a ship was physically possible in this period). It is much more likely that the ships were about 40-60 metres in length (still a very impressive size), in fact the Chinese built a replica ship for their museum which had a length of about 63 metres.

Zheng’s fleet collected much wealth in tribute, including some exotic gifts such as African animals such as Giraffes, the fleet also served a military purpose, as they carried a small army of an estimated 15-40 thousand men, which was often used to cow those that were not intimidated. Zheng’s voyages took place between 1405-1433, but after the Yongle Emperor died, Zheng slowly fell out of favour and the voyages ended in 1433. Not log after that, the Treasure fleet was burned, and China largely ended her foray into becoming a world-class naval power.

On to America, it has been speculated (emphasis on speculated) by certain people that one of Zheng He’s voyages was to America. This is a view espoused most popularly, by retired Submarine commander, Gavin Menzies, his argument is that China had the time, money etc to develop such a fleet, and he therefore speculates that China discovered America before Europeans, however his views basically amount to conspiracy theories, most serious academics have dismissed his claims, as Menzies produces very little evidence. His argument basically amounts to ‘Why not?’. Thus whilst I’d argue it’s true that Zheng’s voyages were remarkable, and their relatively recent rediscovery breaks certain perceptions of Ming China, it is probably untrue that he ever reached America, or Australia or New Zealand; there is simply little if no evidence apart from conspiracy theories. His voyages are very impressive by medieval maritime standards nonetheless.

Edit: After the destruction of Zheng’s fleet; the Ming Dynasty no longer had the desire to invest heavily in a world-class navy; they were instead much more concerned with internal struggles and controlling/ expanding their land borders, particularly in the South-East and in the North. As far as I know, interest in constructing a powerful navy would not take place in China until the late 19th century, during which China constructed several semi-modernised fleets which would ultimately lose to the Imperial Japanese Fleet in the first Sino-Japanese War (Despite how one-sided the conflict appears in hindsight, the Chinese fleet did pose a potential challenge to the IJN, and many in the West did not know initially which fleet would come out on top).

Edit 2: Wow, I didn't expect such a response, I'll attempt to answer questions to the best of my knowledge, using the sources in front of me ('China and the Oceans in the Early Ming Dynasty 1405-1433' by Edward Dryer, 'When China Ruled the Seas: The Treasure Fleet of the Dragon Throne, 1405-1433' by Louise Levathes and Sextants of Beijing). I apologise for delays in answering questions, I’ll try and get round to all of them, I want to make sure they are backed up by several sources which will take a bit more time for answers.

Main source I used: ‘The Sextants of Beijing’ by Joanna Wayley-Cohen

Apologies if the answer is a bit long-winded I intended it as a correction to the previous comment

Here's a short version in no particular order:

1. China and the Americas are WAY farther than Europe and the Americas. It's roughly an extra three thousand kilometers each way as the crow flies compared to the Columbus voyage, and more than double the distance Vikings had to go, assuming they're landing on the continental north america somewhere outside the northernmost part of Alaska, which is still a greater distance.
2. Any hypothetical Chinese voyage absolutely isn't going as the crow flies. It's far safer to try resupplying along the way and if you somehow knew every island and territory you'd probably wind up going south through all of the islands in Southeast Asia and beyond.
3. If you did do that prepared to get caught in a quagmire of geopolitics because all of those islands near to you have a king, emperor, or sultan. Some of them may not be friendly, or in a good place, and likely won't play along with your boldly going.
4. Once you've made contact assuming you don't die of a contracted disease, go native, or die fighting natives, you have to sail all the way back the same distance and deal with yet more problems, both abroad and whatever is happening domestically as you get home.

Intercontinental sailing, as it turns out, is complicated. There's a reason European sailors went all the way around Africa or overland to reach Asia for so long even though they knew the world was round and there was a big ocean on both ends.

However I would be remiss to leave out that the myth of a Chinese discovery has been gaining ground in junk history circles. In the Pacific Northwest people love to point at the occasional chinese coin dug up and fished and claim "the truth", but those coins are generally dated for long after Columbus and attributed to Chinese workers during the gold rush. A few people will claim access to some mythical Chinese wreck full of lost treasure, but they invariably refuse to let anyone else see it or give any kind of interview. Some people like to claim the Zuni tribe is somehow related to the Japanese people and Zheng He's voyage simultaneously and claim there's physical evidence they're related, as well as linguistic commonalities. But those physical characteristics are junk science like bone dents and the linguistics obviously hold no water since the Zuni language is still fairly widespread with many thousands of speakers, and none of them have ever corroborated that claim or verified it with any kind of testing.

There are some claims that an errant boat could be carried by currents from Asia and wind up in the Americas. This is something we've seen happen with sailors and life rafts from time to time, so it can't be ruled out as impossible. However, a ship in such conditions probably wouldn't be able to return home or establish a real colony lasting colony.

I would like to add that these voyages were explicitly political in purpose, not exploratory, and that they were designed to bolster the legitimacy of the emperor who ordered them, the Yongle emperor (r. 1402-24). Zheng He had been a loyal subordinate of the Yongle emperor for many years prior to Yongle's seizing of the throne.

I say "seize" for good reason. A bit of background is helpful to understand the intent behind Zheng He's voyages. When the Ming founder (the Hongwu emperor) died in 1398, his first three sons had already predeceased him. Because rules of primogeniture defined the imperial succession, Hongwu's very capable and middle-aged fourth son, Zhu Di, was not first in line to become emperor. That honor went to Hongwu's grandson by the first (deceased) son, who became the Jianwen emperor. The Jianwen emperor was young, inexperienced, and relied heavily on his officials. Zhu Di was older, an experienced military commander and political leader, charismatic, and had a strong base of power in Beijing. Zhu Di may have been a bit upset about this obvious mismatch in capability, and may have long harbored the notion that his father should have named him as the successor. Within a year of Jianwen's ascension to the throne, Zhu Di launched a rebellion (on the pretext of saving Jianwen from corrupt officials), which quickly became a three-year civil war. Zhu Di won the war, and conquered the capital at Nanjing, but in the confusion of Nanjing's fall the Jianwen emperor disappeared and probably died in a palace fire.

So Zhu Di, now the Yongle emperor, had a legitimacy crisis. He had violated the most basic of Confucian principles by committing nepoticide and taking the throne by force. He wished to demonstrate to East Asia and the rest of the known world that he was truly the Son of Heaven, and that the Ming empire was at the center of All Under Heaven. The maritime voyages of Zheng He, which Yongle ordered, were intended to showcase these claims. Also, Yongle saw himself as another Khubilai Khan (the fifth Mongol kaghan who also founded the Yuan Dynasty and ruled both as a Mongol kaghan and Chinese emperor), and the prestigious memory of the vast Mongol empire was still very strong in the minds of early Ming rulers and statesmen. The voyages, therefore, were also a way for the Ming to stake a claim to the vast ecumene that the Mongols had conquered and connected.

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Some additional reading:

Henry Tsai, Perpetual Happiness: The Ming Emperor Yongle

David Robinson, Martial Spectacles of the Ming Court

Followup question: I know the Chinese were ancient astronomers. Had they invented celestial navigation by this time? (Which I would assume would be required to make it across the Pacific.)