Are there commonly used words to denote different gradations of friendship in English?

I think they might be: 1. Acquaintance 2. Friend/buddy/mate 3. Best friend

I know the last one may not be a true synonym, but it's the best I've got.

Here's a few colloquialisms to add to the spectrum:

• Friends with benefits: A sexual or near-sexual and emotional relationship between two people who don't expect or demand to share a formal romantic relationship.
• Frenemy: A portmanteau of the words fr(iend) and enemy, the term frenemy refers to someone who pretends to be a friend but actually is an enemy—a proverbial wolf in sheep's clothing in the world of friendships.
• BFF: (Best Friend Forever) Slang used primarily in the USA by teenage and young adult women to describe a girl friend or close best friend.
• Brother from another mother: a good friend that is more like a brother.
• Wingman: a friend (male) who helps you find a sexual partner for the night by speaking highly of you.

^{sources: Wikipedia and The Online Slang Dictionary}

Plus a whole host of nuanced urban slang words used for friends such as cuz, homey, dawg, peeps, etc.

Here is my list of generic words (possibly colloquial but not so recently mainstream) in the friend/enemy scale:

• enemy
• adversary
• stranger
• passing acquaintance
• acquaintance
• associate
• colleague
• ally
• comrade
• companion
• friend
• pal/buddy/mate/chum (very colloquial/somewhat male directed)
• good friend
• best friend

Of course, some of those in the middle have connotations that are not necessarily 'friend-like', but they fill out the interior of the scale.

As I think Denis Leary once said,

A friend is someone who will help you move. A true friend is someone who will help you move a body.

Despite what others have said, an acquaintance is not really a friend.