What does "clockwise" mean, exactly? I am in the middle of a discussion with a friend about the meaning of the term "clockwise".
Wikipedia indicates that a clockwise rotation goes as top-right-down-left.
However, my friend argues that "the clock is facing opposite to you. So if you rotate top-right-down-left facing the side the clock is facing then it is anti-clockwise."
Our question, therefore, is

does "clockwise" mean top-right-down-left or left-down-right-top?

 A: The clockwise direction is normally defined by the right hand grip rule.

When your thumb is pointing away from you, your fingers are curled clockwise. So when you look at a clock the axis of rotation is away from you through the clock.
I'd guess the downvotes are because people believe your question is not physics related, but in fact this rule is how you determine the direction of the angular momentum vector, so there is a connection with physics.
A: Your confusion arises because the term (anti)clockwise, when used by itself, is ambiguous, and should always be used with a statement like "as seen from the top" (unless that is absolutely obvious$^1$). The reason for this is that "clockwise" defines a direction of rotation within a plane, but does not specify which side the plane is observed from. (In more technical language, "clockwise" defines an order for two basis vectors for the plane, but does not specify a sign for the normal.) If you take a transparent clock mounted on a glass window, it will rotate clockwise or anticlockwise depending on what side of the window you're standing.
That said, a statement like "clockwise as seen from the top" is not ambiguous. This refers to the direction of rotation of the hands of a (normal, non-transparent) clock as you observe them. It coincides with the direction your fingers point to when you put your left hand in front of you with your thumb pointing towards you. In your terms, 

clockwise means "top-right-down-left".

Additionally, if the question is purely about a planar problem (i.e. when doing 2D analytical geometry), then there is a canonical way to understand the term. Put the page on the wall so you can read it, next to a clock whose face you can see; its hands will move clockwise. This is equivalent to setting the plane normal "up", towards you. Put your left hand on the page with your thumb toward you, and your fingers will be clockwise.
It must also be noted that the "positive" direction of rotation is usually defined to be anticlockwise. Given a directed axis, put your right hand with your thumb pointing along the axis in the given direction, and then your fingers will point to the "positive" direction of rotation.
$^1$One example of this is hurricanes, which are said to rotate (anti)clockwise in the (northern) southern hemisphere. Here of course you're supposed to look at them from above (e.g. from space). If you see one from the ground (preferably from the eye, and from inside a well-protected building) the rotation will be inverted.
A: Apparently, there seems to be no agreed upon definition of clockwise in 3D space - but an a clear agreement regarding the meaning in a plane "seen from above"
As I am starting to answer, I have no idea what is the correct answer,
if any. It is the "if any" that gets me started on this topic. I am
certainly not an expert, I never could tell my left from my right,
especially after having physics classes, but I am looking for the
available evidence.
To be honest I got actually very confused by John Rennie's answers, because I did not see the need to introduce yet another device, which you have to orient like the first, but not so naturally ... At least there is only one kind of clocks (except for funny souvenirs), while there are two kinds of hands to add to confusion. Also there is only one natural way to look at a clock, while there is no specific or normalized way to look at your hand, though having the thumb first is the natural position ... !
The word "clockwise" is used more than 800 times on physics.stackexchange.com
according to the very approximative figures of google (I used:
clockwise site:physics.stackexchange.com). This adverb must have some
relevance to physics.
When searching the web, or wikipedia, for a formal definition of
(counter-)clockwise in 3D, there seems to be none to be found. the answer by @Emilio is close to mine as he does seem to consider that the adverb "clockwise" has a well defined meaning without further precision.
The Internet being somewhat too large a place for me, I looked at
the uses of this adverb on physics.stackexchange.com. I even limited
myself to the first twenty answers from Google, as this takes
considerable time. Be my guest to do more.
Many uses are simply informal. Wich way does this or that turn when
you (un)screw or look at its rotation (like a clock). Bicycle pedals
is a good example. In many other cases, it is only used to indicate
changes in the sign of angular velocity, with corresponding changes in
the sign of other quantities, without any precise chiral information.
Note, for example, qthat when talking of hurricanes, it is always rotation as seen from above, like a clock seen from the front.
In Vector Nature Of Angular Velocity, the use is
clearly a 2D planar use, about a plane seen from above. The vector
considered does follow the right-hand rule. But it is not construed as
a definition of clock-wise rotation in 3-D.
The same is true in Alkali atom in oscilating electromagnetic field. Indeed, it talks of counter-clockwise rotation as it look at the right
hand from the end of the thumb.
In the answer to question
What is negative angular acceleration?, things are a bit
confused as it mentions the right hand rule, but seems to describe the
opposite.  Then it also defines clockwise as the positive rotation,
while it is usually considered negative in mathematics (well, it did
in my school times, but I would not know the current fads). Again,
this is not construed as a definition of clockwise in 3D.
Question Lenz' law versus $-\frac{d\Omega}{dt}$ uses
"clockwise (using right-hand notation)". The answer too makes
explicit reference to the right hand rule to define clockwise.
One answer to Applying the right-hand rule for magnetic forces
explicitly uses the right hand rule, and explains that it is useful
because the direction is "counter-clockwise or clockwise depending on
what side of the plane you are looking at".
One thing in common to all the uses where it matters is that the
authors do not seem to think that there is an agreed upon definition
of (counter-)clockwise in 3D which they can rely on for unambiguous
communication.
The other thing in common (except maybe for the third)
is that seem all to understand clockwise as "like a clock seen from above", when you use the right hand rule to translate what is being said.
Based of this evidence, admittedly needing more investigation,
my own temporary conclusion is that, unless someone can point to an
"official" and agreed upon definition of the concept of clockwise rotation in 3D,
it is probably inappropriate to assume such a definition.
The other conclusion is that there is clear consensus that clockwise means, in a plane, like a clock arms in the same plane assuming the clock is facing you.
An interesting aspect of it is that the word (counter-)clockwise seems
very much used because it has intuitive meaning, apparently more than
the right-hand rule. But it has sometimes to be defined more
precisely, by the right-hand rule or equivalent means (you look at a
clock in the XY plane from the positive Z ...).
Another clear conclusion, at least to me, is that it is an
appropriate question, and it should be upvoted. It may not
be physics in the most basic sense, but it is obviously a tool for the
physicist, and without proper tools technicians cannot work.
