What's the difference?

Hey guys I have a question for you.
What's the difference between " a few" and " few"?
I asked my English teacher this question last year, but his answer didn't make any sense to me.

Usually "a few" means "some" whereas just "few" means "not many" but be careful of "quite a few", "a fair few" or "a good few" which mean "lots". Perhaps the dictionary would make things clearer.

http://www.dictionary.cambridge.org/results.asp?searchword=few&image.x=32&image.y=10

few (SOME) [Show phonetics]
determiner, pronoun
1 a few some, or a small number of something:
I need to get a few things in town.
There are a few cakes left over from the party.
We've been having a few problems with the new computer.
If you can't fit all the cases in your car, I can take a few in mine.
"How many potatoes do you want?" "Oh, just a few, please."
NOTE: a few is used with countable nouns. Compare little (small).

2 used in expressions such as 'quite a few' or 'a good few' to mean quite a large number:
I know quite a few people who've had the same problem.
Lots of people at the club are under twenty, but there are a good few who aren't.

http://www.dictionary.cambridge.org/define.asp?key=28586&dict=CALD

few (NOT MANY) [Show phonetics]
determiner, pronoun, noun, adjective
a small number, not many or not enough:
It was embarrassing how few people attended the party.
He is among the few people I can trust.
Very few people can afford to pay those prices.
We leave for France in a few days.
Few of the children can read or write yet.
Few things in this world give me more pleasure than a long bath.
Fewer people smoke these days than used to.
We get few complaints.
According to the survey, as few as 10% of us are happy with our jobs.
The benefits of this scheme are few.
NOTE: few is used with countable nouns. Compare little (not enough).

http://www.dictionary.cambridge.org/define.asp?key=28589&dict=CALD