In the most simple terms flushing means you are actively washing the old fluid out using some sort of other liquid (such as a detergent or solvent) rather than passively draining which is just allowing as much as you can to flow out of the system unaided.
Does flushing require a special machine?
In engine oil terms a flush is done by adding a special detergent to the existing oil and then idling the engine for ~10-15 mins and the engine effectively "pumps" the flush around the oil system cleaning as it goes. You then turn the engine off and drain it as normal.
flushing your engine or transmission oil can sometimes cause damage on old vehicles which didn't have regular oil changes done to them
Yes, unfortunately this a real phenomenon, sometimes the sludge/deposits that the flush clears away are actually masking a degraded seal. Meaning that while it's not done damage it's exposed existing damage and made it start causing leaks and similar problems.
The older and more poorly maintained the engine the more likely it is for this to be the case. If you suspect this is a possibility on an older car that you're just planning on using until it expires this means flushing is probably going to be a Bad Thing(TM) as tracking down and replacing the seals could start to get expensive. If it's an engine you are aiming to have last as long as possible and are putting the time, money and effort into bringing to peak condition then it can be good - you can address any issues the flush reveals and drive on safe in the knowledge that you are less likely to have a hidden issue waiting to bite.
Transmission fluid flushes are a bit more complicated - here a machine is often involved (which means finding a garage that has one) so it's going to end up costing you more. Here there are risks also - different to the engine but probably more likely. Depending on the transmission it's possible that a flush can push fluid into areas or in directions that the transmission wasn't designed for, and that can cause damage. Some manufacturers specifically warn against doing flushes for this reason.
The nature of the way a transmission functions also means there's a likelihood of small bits of debris floating around in the fluid from wear and tear - and a flush can push this debris in to places where it can do significant damage.
Is it correct that when you drain you only remove half the oil?
It varies - on most engines if a oil change is done properly you'll remove more than half, but it's unlikely to be all the oil, and if you do a couple of changes back-to-back (with about a 15 minute idle in between to circulate the oil) you'll get more or less all of it, but that's rarely necessary IMO unless you have particular reason to suspect the oil is compromised in someway.
For a transmission change this is more of a thing - many autoboxes will only drop around 30-33% of the fluid with a pan drop, which can be worked around with repeated fluid drains so that over time you effectively dilute the "old" fluid and replace it with new - doing a drain 2-3 times with some driving in between should pretty much do it.