We use the terms range, register, ambitus and tessitura with regard to how high and low an instrument or vocalist can go. I'm not at all sure how or where the terms cross over, and indeed whether they all refer to voice, instrument, or music. There may well be other terms that come into play. Please enlighten me!
Interesting question; I didn't realize my own confusion until you pointed it out.
According to The Harvard Dictionary of Music:
Ambitus. The range of pitches employed in a melody or voice.
Range. The span of pitches between highest and lowest of an instrument, voice, or part; also compass. See also Tessitura.
Register. A specific segment of the total range of pitches available to a voice, instrument, or composition. It may often be described loosely simply as high, low, etc. For the nomenclature of registers of the voice, see Voice.
Tessitura. The particular range of a part (especially a vocal part) that is most consistently exploited, as opposed to the total range or compass of such a part. Thus, a soprano may have a high or a low tessitura.
Voice. (excerpts) Voices of similar range may be unlike in tessitura---that part of the range that is most comfortable for the singer and sounds best. ... In singing, register refers to sthe eries of pitches that are of like tonal character because of a consistent vocal production.
Part of the difficulty may stem from the use of the term "voice." Occasionally it means someone's voice, other times it means a particular music line. But as far as I can tell, more user-friendly definitions are:
Ambitus. The span of pitches used by a single line (vocal or otherwise) in a musical composition.
Range. The entire span of pitches that can be produced by a single line (vocal or otherwise). This is not limited to a particular composition, and is synonymous with compass.
Register. A segment of someone's entire range (vocal or otherwise). This is a general term and is not limited to a particular composition.
Tessitura. A segment of someone's entire range (vocal or otherwise) that is most consistently used, either by a particular vocal part or within a particular composition. This is a more specific term than register because it can be limited to a particular composition, but it doesn't have to be.
(Note that these are just my understandings of these definitions based off of the Harvard Dictionary; I don't guarantee accuracy like I do most of my other answers!)