Depending on the job you are printing there are several different options as to how to print them.
A digital job, when it first came out the resolution was never considered as good as press, however now on a good quality digital machine most people can not tell that it the job was not run on a press. When digital first started, you know how sometimes on your printer at home you have streaks through the ink, that was not uncommon when digital started, however now it has become a much better process. A digital machine is more cost effective on small quantities and is not cost effective on large quantities.
The benefit of having press and digital is that when you have jobs that are on the cusp of size, you can have it quoted both ways so that you can tell the difference.
A press is the highest quality of print available. There are presses that can do one colour only, and there are presses that can now print up to 12 colours in one line. A press is used for larger jobs. What used to be is that if you had a 4 or 5 colour press you would run it through with the 4 colours and maybe a coating or a spot colour, pull the job, let it dry and if it was double sided flip the job and re run it, however now with 10 or more colour presses there is an option for double sided jobs when they go in to print on the one side, the machine automatically flips it and prints the other side. This can help save time and money.
There is also a web press for huge jobs, and this is how news papers are printed.
So when you are now talking to your printer, make sure your job is being done on the right machine.