It's the transitive verb "to thank". Obrigado is translated as "thank you" but literally it comes from "(I am) obligated (to you)".
As pointed out by bigglesworth elsewhere in this thread you can translate "obrigado" by "much obliged" instead to make it less confusing. In this case "thank you" would be very literally "eu agradeço a você".
Dungna93 I found your post a very helpful way to remember Danmoller's tip that agradecer takes the preposition "a". If it's translated as "to say thanks" as you suggest, it makes it much easier to remember that it requires the preposition "to" = "a" in Portuguese. Eu agradeço a você por isso :-)
Important note: O aluno agradece a professora = The student thanks the teacher BUT O aluno agradece À professora = The student thanks TO the teacher. À = contraction of the preposition "a" with the article "a", to avoid 2 "a"s together. Á = this just doesn't exist, so, never write it, it's a common mistake that we Brazilians make. Ex.: Eu chego às 10 horas (correct) Eu chego ás 10 horas (totally incorrect) Obs.: "ás" can be used, but has the meaning of "ace" Ex.: I don't have an ace (eu não tenho um ás) He is the basketball ace (ele é o ás do basquete)