Mon vieux is a way to refer to a dear friend. Its English equivalent would be “old chum”, “old pal”, or “dear friend”.
As with the English expression “old chum”, vieux doesn’t refer to the age of the friend, but rather the amount of time you’ve been friends with him. So, with that in mind, this isn’t something you’d call an acquaintance or someone you’ve recently become friends with.
Mon vieux is a bit old-fashioned but still used in everyday informal or neutral French. It’s not used in formal or professional situations, though. It would also probably seem weird if you were said it around young people who were using a lot of slang.
There is an equivalent for a female friend, ma veille, but this is used far more rarely, possibly at least in part because referring to a woman as veille could seem like an insult (even though it shouldn’t be!).
Speaking of which, be careful to always use this with the possessive pronoun mon (or ma for veille) and not le (or la for vielle), since le vieux/la veille means something like “the old fart/the old bag/the old geezer” – definitely not a nice way to refer to a friend!