It is worth noting that there is also a slight difference in usage of these terms in American and British English. Also, the word “mate” is used for ‘friend’ in the UK. “I am having dinner with my best mate tonight.”

Buddy = friend Pal = friend Chap = a man (but a very old expression) Fellow = a man (but also an old expression) I use "buddy" often, but I don't usually use the other 3 words.

I would emphasize one small difference between a friend and a pal or a mate. When you say "the best friend" you mean a person who is the closest person to you. He or she knows everything about you. You spend all time together. When you say "a friend " you mean that he or she is still a person who is close to you. But not the closest. You have common interests and often spend time together. But when you say "a mate" or "a pal" or "a buddy" you don't mean anything these. You can say "a mate" to anybody whom you know and relate to him or her positively. You don't insert a meaning of the best friendship into this word. Meanwhile you always can refer even to your best friend as "body" or "mate" or "dude", you will just not vocalize in that moment that he or she is a very close or important for you person. Because people don't need to express their feelings constantly. Hope it helps and people will agree with my explanation.

I agree with what Ian and Collin have said. Chap is more common in British English. I don't think "fellow" is such an old-fashioned word (but maybe I'm old). Note that "buddy" and "pal" can be used in place of someone's name, especially if that person is a friend. Example: "Hey, buddy, how are you?" But also, if you don't know the person, both "buddy" and "pal", but especially "pal" used as a name, can sometimes sound a little unfriendly or aggressive. "Hey, pal, just mind your own business."