Impressment in the Royal Navy has a longer history than it's more popular state as a cause of the War of 1812, large scale modern impressment can be dated back to the 1690s. It tended to be at it's highest points in wartime, naturally, as the Royal Navy needed more sailors and seamen than during peacetime. Sailing in the merchant marine gave better wages, conditions, and protection than in military service, and so the call to volunteer was generally met with negative response. Therefore, RN impressment tended to be located in port cities and largely looked to target skilled seamen (as opposed to the popular image of press gangs simply getting any poor drunken body on a ship) to work on warships.

In the post-American Revolutionary period and the Napoleonic War period, the need for more men at sea again motioned the Royal Navy to look to impressing merchant sailors and civilians into service. An added factor was the status of American independence. Impressment of foreigners was nothing really new, Denver Brunsman puts the figure of foreign citizens in British service in 1805 at around 12 percent. However, the want of British merchant sailors and Royal Navy deserters to escape the press gang led to a flight of sorts to the young American merchant marine. The linguistic and aesthetic similarities of the two nations' peoples provided an easy disguise and transition for British marine runaways, while also giving the hope of protection under an independent sovereign State's flag. The American merchant marine benefited from an influx of skilled and culturally similar seamen.

The Royal Navy was understandably not happy with it's sailors and seamen avoiding the possibility of service, along with it's need of manpower, so it declared it's own right to board foreign (typically American) vessels, and collect and impress what it considered to be British citizens and deserters. The debated definitions of what the United States and United Kingdom considered 'American' and 'British', as well as the zeal British captains took in reclaiming British seamen, meant that inevitably American mariners were caught up in impressment. This was not helped by the vague orders and crude ways of identifying British deserters given to British captains to reclaim seamen, unintentional or malicious "overlookings" of American status, and practices by actual British deserters to exploit systems to protect American seamen (taking up fake identification, fraudulent papers, etc.). I discuss that a bit more in the linked post.

So what was the experience of a captured American seaman like? Unfortunately this sort of falls into the "depends on the sailor" category. After all, with some ten thousand Americans pressed (a figure again given by Denver Brunsman), we will inevitably have common themes of their lives in service and also unique events, fates, and ultimate experiences. I can, however, link you to a nice collection of contemporary documents relating to British impressment of American seamen here, including letters and accounts. Be a bit wary, as any historian should, in reading these. Impressment in the United States, even ranging as far back as in it's colonial period, had been a very passionate issue. It's very possible, even likely, that some exaggerations may be present to continue demonizing the Royal Navy.

How did they manage to 'press' so many unwilling people into service? Wouldn't there have been a rebellion?

This is an interesting factor that I'll answer through, yet again, Denver Brunsman (whose book The Evil Necessity: British Naval Impressment in the Eighteenth Century Atlantic World is one of my main sources on the topic, and an excellent book for those looking for more information about the history of Royal Navy impressment). Surprisingly, he gives a figure of 70% of desertion by pressed sailors in their first year onboard, afterwards it falls dramatically to under ten. He gives some factors into this remarkable fall in percentage by giving examples of how pressed sailors could adjust and come to terms, or acceptance, of their position. After all, the majority of pressed sailors were skilled seamen before service, and continued to be so in service. The large cut in pay, harsher discipline, and longer periods at sea would encourage a lot of pressed sailors to escape. The ones who stayed, however, often looked to use their situation to their own advantages, looking to succeed in a mixed-meritocratic environment as ship's company and their skills/experience to earn better conditions onboard. As well, in contrast to, say, slavery, pressed sailors could at least "wait it out" and be released on conclusion of the war that they were pressed to serve in. We do have accounts of pressed seamen who succeeded and advanced in their service.

Resistance came in many forms from many different groups, the pressed seamen, port citizens, etc. The most common was 'passive' forms of resistance, and quite simple in concept: desertion and avoidance of press gangs. Pulling into port especially, pressed sailors who wanted out would do their best to disappear into the town/city they pulled into. Avoidance would be done by sailors on land or on sea by a variety of methods, disguise, assuming of false identities, taking up fake jobs on land to avoid their status as a skilled sailor, etc.

Violent resistance onboard vessels is a lot more rare in occurrence, things like mutiny and outright rebellion by pressed sailors aren't well attested to. It's probably that the idea of sabotaging the vessel you're on is not in your best interest, or in the interest of other crew members, pressed or volunteer. As well, the systems of discipline and possible "bright side" outlook of some pressed sailors that I mentioned previously dissuaded thoughts of violent resistance on the vessel. A lot of violent protest came from port cities and their people, looking to fight away the press gangs and keep the Royal Navy from taking away their men. In the metaphorical limelight of this fight are the American colonies. Noticeably, we have the Knowles Riot of 1747 in Boston as a large and fiery example of American resistance to impressment, when an Admiral Charles Knowles looked to press Massachusetts men into his fleet, the citizens revolted, abducted some of his officers, and held them hostage in exchange for the promise of keeping Massachusetts residents from service. Boston reinforced this promise in 1768 with the Liberty riot, and impressment in the American colonies was largely removed to placate an agitated populace on the Eastern seaboard that made a lot of it's living off it's seamen.

EDIT: My recommended sources to go to for more on impressment and also it's practice against American citizens:

Free Trade and Sailors' Rights in the War of 1812 by Paul A. Gilje

Sweatshops at Sea: Merchant Seamen in the World's First Globalized Industry by Leon Fink

The Evil Necessity: British Naval Impressment in the Eighteenth-century Atlantic World by Denver Alexander Brunsman

Six Frigates: The Epic History of the founding of the US Navy by Ian W. Toll