By degree we mean a 3 or 4 year BA degree; Associate Degrees, Foundation Degrees and other shorter course qualifications are not accepted as degrees in most countries.
The degree is usually required for visa or work permit purposes. In most cases the subject of the degree is not important and you will find teachers of English with degrees in everything from Medieval English Literature to Mechanical Engineering to IT to Sociology.
Having said this, to help get the job it's obviously better to be English or language related, for example a degree in English Literature.
If you are thinking about which degree to take, you might want to think about taking something which interests you! There's little point in spending 3 or 4 years studying something you dislike when, at the end of the day, you would be in the same position as if you'd studied something you enjoyed.
Some universities offer degrees in TESOL or similar. Bear in mind that if decide at a later date to leave the TEFL industry and enter mainstream employment back home then most employers will not know what this is. A degree in English is sometimes better since most people will be able to recognize this.
Masters Degree.- for a further qualification if you are heading towards a career in TEFL.
TEFL without a Degree - if you do not possess a degree and wish to teach.