How seriously do professors take teaching evaluations? Do they ever know who writes them? How seriously do departments take teaching evaluations? If a professor knows which student wrote a particular evaluation....would they be biased (e.g. be nicer, etc...)?
Please include in your answer what country you are in and a short description of what type of school you are at, and what you do there.  Answers presumably vary between research universities and four-year colleges, for example.  Also, professors and students could reasonably have different access to information about how teaching evaluations are used.
 A: To answer your first question: teaching evaluations are supposedly anonymous.  However, two experiences I've had are relevant:


*

*at some universities, including the university at which I am currently a graduate student, the students are invited to write comments by hand on the evaluation form.  These comments are sometimes photocopied and distributed to the instructor along with a summary of the numerical evaluations.  In a small class it's not impossible that the instructor would recognize the students' handwriting, and so I know who wrote comments about some of my courses.  This is of course not an issue if evaluations are done online, which increasingly many institutions (including mine) do now.

*as an undergraduate I took a course that was being offered for the first time.  This course was somewhat experimental in nature and so the professor in charge handed out a mid-semester course evaluation form.  I wrote some comments in which I inadvertently gave enough information that it was possible for the professor to identify me.  Since it was clear that I was frustrated with some aspects of the course he had me meet with him.  (In the end this worked out well, as this professor wrote me a recommendation for graduate school.)
A: You have excellent answers concerning anonymity.  Regarding how seriously they are taken, it varies widely.  In the faculties I have been involved in (except, I think, Yale, but my involvement was minimal there), the evaluation process was an informal, department-internal matter, and so how seriously professors take their evaluations is a matter of (i) department politics, and how secure the professor feels in the department, and (ii) conscientiousness. 
I have known a professor who simply said that he did not care what his students thought of his courses, he knew what he was going to teach, and he wasn't going to hold any hands.  His evaluations were awful, I think mostly because his course was much tougher than the other courses, and as one of the strongest research professors in his department, this did not cause him any trouble that really bothered him.  I should say, he has an excellent presentation style, very clear, very logical presentation, and I respected his stance.
A: Read Peter Sacks's book Generation X Goes to College (Open Court, Chicago, 1996), at least through chapter 9.  Case study of negative correlation between eval scores and teaching quality, and a how-to manual on manipulating your ratings.  Despite coming from a different field, Sacks's story in chapter 2 about his threatening student "Pete" almost exactly mirrored some of my experiences pre-tenure.
A: I'm not sure whether I should really post these few thoughts of mine because they do not directly address the question as it was asked but every time somebody starts talking about students evaluations I feel a strong urge to say something like that, so why not now and here? As usual, I believe I'm guilty of violating the "no discussions on MO" rule, so feel free to downvote. 
I find the current system of term end student evaluations extremely counterproductive. On one hand, most students are just plainly incompetent to answer many of the typical questions (like "rate your professor's knowledge of the subject"). On the other hand, it creates an unhealthy situation when instead of getting a constructive criticism during the course when the adjustments can easily be made, we just get emotional comments on our performance when it is too late to do anything and more often than not in the format that is impossible to interpret in any reasonable way. Out of each 50 evaluations I got only 3 or 4 contained something that I could use to adjust my teaching. The rest either contained pure rating numbers that didn't tell me anything I hadn't known myself already (yes, I did know if the topic was hard or easy, I did know whether I prepared my lectures carefully or just improvised, etc.) or the comments like "This is the best teacher ever!" or "He should be fired immediately!" (sometimes I got both in the same class), which also carried no real content whatsoever. 
I agree that it is a great idea to get the feedback from the students and I admit that some students may be afraid to express themselves openly, but it would be much better just to have a small forum type website for each course where every enrolled student would be able to publicly post any concerns or comments  about the course in a free style format as soon as they arise with or without signing it (both options should be available) and the teacher would be able to read those and to respond and decide whether any action/adjustment is due. Such records would also tell much more about everybody's teaching to any promotion committee or whoever else who might have a real interest in what is going on. 
Technically it is not so hard but I've never seen anything like that done. Instead, we have that system of meaningless average numbers of subjective ratings of rather poorly defined things. Indeed, what is my "being organized", or "preparedness", or "willingness to help"? I believe I can read and speak basic English but I'm just unable to assign any meanings to such general words, much less to assign a numerical value to the qualities they are assumed to represent. I understand the sentences like "You never returned homework #2" or "Your lecture on continuity was very clear", but not the sentence "Your overall preparedness was 5 on the scale from 1 to 10". 
The most unpleasant thing is that the administration tends to take those numbers rather seriously sometimes. I don't care: I've got my tenure and my salary is high enough but it is a pity to see young people who are afraid to teach $\varepsilon-\delta$ in calculus courses because it may irritate the students and result in bad evaluations.  
As to the requested background information, I'm currently a full professor of mathematics at University of Wisconsin-Madison but I spent more than 10 years at Michigan State University before that. In Michigan we were required to do evaluations for every course taught; Wisconsin requires just to do them "now and then" (not less than one in 3 semesters or something like that; I still have to figure the exact rule out). I wouldn't say they were taken very seriously by the faculty in either place but I cannot speak for everyone. I have never tried to figure out who wrote what for the lack of time and curiosity for such things but I believe Leonid that it may be possible to do in a small size class.   
A: To answer the second question: professors at most universities these days take teaching evaluations fairly seriously -- not necessarily one by one (although we are human beings and a piece of seemingly undeserved criticism can stick in our craw as much as anyone else, if not more so because there is no other aspect of our job in which we are as openly criticized) but certainly on average.  Probably 95% of professors are or were in the situation where having numerical scores below a certain threshold creates trouble for tenure and promotion cases.  (And once you get into the habit of caring about evaluations, it's hard to break.  A colleague of mine is a full professor but is just as concerned that her evaluations stay high as I -- an untenured professor -- am.)  
Most departments these days have a sentiment that although great evaluations are not necessarily highly correlated with great teaching (in part because it is not so clear what constitutes great teaching!), really poor evaluations probably mean that the instructor is doing something wrong, not necessarily content-wise but in the way s/he is relating to the students. 
