Many teaching institutions request a TESOL or other ESL teaching qualification in addition to a degree. This is because they want to be sure that the person they hire has received professional training in the area in which s/he will be employed, namely teaching English as a Second Language.
There are no standard requirements for TESOL. It depends on where you want to teach and at what level. Many European countries require you to have a TESOL, TESL or TEFL certificate to teach. Other countries may simply prefer those who have certification.