There is some debate about the correct usage of the words obrigado and obrigada. The usage indicated above is by far the most common way the words are used, that is, men always say obrigado and women always say obrigada. This usage implies that the word is being used as an adjective to describe the one speaking (literally meaning ‘obliged’).
Technically , it can be argued that the word obrigado, when used on its own to say ‘thank you’, is an interjection, not an adjective (in the same way that the word ‘hello’ is an interjection). Under this school of thought, both men and women should say obrigado regardless of whether they are addressing a man or a woman. Whilst this is probably the ‘correct’ usage, it is hardly ever encountered. Of course, if you are male, it doesn’t really matter as you would say obrigado anyway.
In some regions, particularly the Algarve, it is common for both men and women to use both obrigado and obrigada - switching between them depending on the gender of the person they are talking to. However, this usage does not seem to have any rational technical explanation! I would therefore recommend against that usage unless you happen to live in a region where the locals would be offended if you addressed them differently.