Hola Javier! 'Colleagues' are people you work with. Most people use that word in a formal capacity because you could always say 'coworker' and it means the same thing but is a bit more casual. 'Mate' is a difficult one...in America we don't really use that word. It is more often used in Australia or England where it just means 'friend'. 'Mate' is also at the end of some other words that we do use in English like 'roommate', or 'housemate' those both mean people you live with but if you put 'mate' on the end of another word it just means person you do that action with. 'Partner' had different meanings depending on the situation. Someone can have a business partner meaning a person they own a business with. People don't usually use 'partner' to describe the person they are dating unless they don't like the words boyfriend and girlfriend. However, sometimes a person will say 'partner' when they aren't married but don't consider themselves to be dating. You have to listen to figure out what this one means in conversation. I hope this helps. Puedo explicar en español si tú necesitas. Seguir aprendiendo :-)

You seem to have got it, basically, but there are some subtleties. A partner could also be a business partner, or a partner in some other enterprise, such as a "lab partner" in a experimental science class. The meaning of "mate" is very context-dependent: "He's gone to the bar for a drink with his mates", "I have a meeting tonight with my study mates", "This bird is nesting with its mate", "The captain gave his mate the order to heave to" (or something like that -- nautical English isn't my linguistic strength!). In the US, "buddy" is often used for the first two of these meanings.