We're painting with a broad brush, but this is still a useful
simplification for the sake of discussion.
These are all kinds of security problems. What's being described is
what kind of privileges the attacker A had on computer
C at the outset, and
what kind of privileges A ended up with on C after the attack.
remote root - Here A starts with no access to
C, and ends up with root access directly; they can do anything
they want on C with no extra steps.
remote nonroot - Here A starts with no access to
C, and ends up with relatively limited access directly; they
can run shell commands for example, but they don't have full control of
the machine. However, remote nonroot allows an attacker to try a
large number of local root exploits to effectively promote this
to remote root - it just takes longer.
local root - A starts with an account on C,
and ends up with full access to C; they can do anything they
want with no extra steps.
local nonroot - A starts with an account on
C, and ends up with greater access to C, but not full,
complete access to C, as would be the case with local
root. This sometimes makes it even easier to get to local
remote DOS - Here A has no account on C at
the outset, and ends up with no greater access. However, A has
prevented users of C from being able to use C for
something it should be able to do. For example, A might cause
a web server daemon to die, preventing users from being able to access
web pages served by C, or A might cause a daemon to
inhale so much virtual memory that C is run out of swap space
and becomes unusable for all purposes until rebooted.
local DOS - Here A has an account on C, and
uses that account to prevent others from being able to access services
provided by C. local DOS attacks are very easy in the
computer world; for example, anyone with an account on a unix or linux
computer can easily run a malloc bomb, which quickly runs the
machine out of virtual memory. Hence, any time you give someone an
account, you are granting a level of trust to that person, assuming they
will act responsibly.