I am writing an essay and I have previously written and submitted an essay to a different professor about a similar subject. Would it be considered plagiarism if I copied my own exact sentences from my previous essay?
Plagiarism and self-plagiarism sound similar, but are importantly distinct things.
Plagiarism is if you use other people's work as your own. You deprive them of credit and claim credit for yourself which you do not deserve.
Self-Plagiarism does not deprive the author (you) from credit, but the offence here is that the the "present-you" claims credit that the "past-you" already collected.
This is an assessment offence because you would get a double mark for a single piece of work. You could, of course, quote yourself (properly cited!), in which case, you would be perfectly fine, but you would only gain marks for new work done, which, in case of a full copy, would still be zero. However, in a sensible school policy it should not be punishable for plagiarism in the conventional sense, since it was you in the first place who wrote the text.
As a student, I have reused software libraries I wrote in other coursework, properly cited, of course. It would be a waste of time to redo mechanical work, and permitted me to get much further. With an essay, nothing stops you, of course, to build upon existing writing (properly cited) to develop your argument further.
Yes, it is called self-plagiarism. This happens when you reuse significant portions of an earlier work without acknowledging this. This can be a major concern if the prior work is copyrighted as it infringes on the rights of the publisher.
If the work comes from any other sources, it is wisest to cite where it came from, even if it came from you.
Terms used are self-plagiarism (see @Darrin Thomas) or autoplagiarism. One of its issues in scientific publishing is "duplicate" or "multiple publications", that brings very little novelty to the subject. I have witnessed the same paper published in 3 or 4 different conferences.
Such behavior are now ruled by scientific societies. For instance at IEEE (note that I am quoting, the original source is under the link):
IEEE Publications has long maintained the policy that verbatim copying of another's work (plagiarism) is unacceptable author conduct. In November 2002, the IEEE Board of Directors approved a new policy on Duplicate Publication and Self-Plagiarism. [...] authors should only submit original work that has neither appeared elsewhere for publication, nor which is under review for another refereed publication. If authors have used their own previously published work(s) as a basis for a new submission, they are required to cite the previous work(s) and very briefly indicate how the new submission offers substantial novel contributions beyond those of the previously published work(s).
One legal reason could be that such societies own some rights regarding the copyright. Not a proof, but in the Wikipedia section on Self-plagiarism, you can find:
In addition there can be a copyright issue if copyright of the prior work has been transferred to another entity
If your work is not published or evaluated, reuse of sentences you have written sounds like a fair use. If submitting an essay induces a legal framework (grade, assignement), you should take some care. You "own exact sentences" are a matter of quantity. At least, it is important that your fairness can not to discussed. Thus, if you mention that you already have worked on a similar topic (with references), you have less chances to be blamed for hiding important information.
As a side note, apart from the time gain, I strongly suggest you to rewrite, without looking too much at the original source, apart to make sure you have not forgotten ideas. Your though has evolved by thinking about it once, and writing it, and reformulating is a good way to improve your line of reasoning and the clarity of your written expression.
If your material is to be published in a scientific conference, journal or as public demonstration of some sort, it would be termed as self-plagiarism. This is when you replicate significant amount of detail from a previously published matter.
I am assuming you are writing an essay to portray your knowledge and understating in the subject. I don't suppose 'self-plagiarism' comes to play here. But it may seem wiser to let the professor know. However, it would best to check the university course guidelines and verify whether you could submit the same essay to a different course as stated.
It may not be plagiarism, but there may be university policies against it. In fact, there may be university policies against using similar essays for two different subjects even if you don't copy phrases exactly.
From Harvard's plagiarism policy:
It is the expectation of every course that all work submitted for a course or for any other academic purpose will have been done solely for that course or for that purpose. If the same or similar work is to be submitted to any other course or used for any other academic purpose within the College, the prior written permission of the instructor must be obtained. If the same or similar work is to be submitted to more than one course or used for more than one academic purpose within the College during the same term, the prior written permission of all instructors involved must be obtained.
Depending on the school's policies, you could get an F in a course for something like this. I doubt the penalties would be as severe as those for plagiarism, but I recommend against doing it without asking permission.
In fact (although this may be due to the fact that I work in science and not the humanities), I would be inclined to be much more forgiving for a student project in the same area as an earlier project of theirs that copied some phrases in the introduction, rather than a project that had essentially the same scientific content as an earlier project, but different wording.