"A few" means "some" in the sense of "not zero".  For example:

Have all the guests gone home?    No, there are a few people left.

Are the tickets all sold out?            No, there are still a few that haven't been sold.

"Few" means "not many".

In spite of the advertising campaign, few people bought the product.

He was a lonely man with no family and few friends.

Compare this last sentence with this:

Do you think that John is lonely?    I think he's fine. He has a few friends.

Can you seen the difference?

He has few friends is negative : it means that he doesn't have many friends, and so he's lonely.

He has a few friends is positive: it means that has some friends and so he isn't completely alone.

Likewise:

I have few problems is a good thing: it means that I don't have many problems.

I have a few problems is a bad thing: it means that that I have a number of problems.

I hope that all makes sense!

A few means a small number of something: three or four, maybe five

Few means "not many" or "almost none" and emphasizes the lack of something.

I have had a few problems with my car.  (Some things have gone wrong, but not many.  But perhaps I am unhappy.)

I have had few problems with my car.   (My car has had almost no problems.  I am happy.)

Few means only a small number.

A few means some. 

Few men know the meaning of respect

There once was a man from Japan who had a pan in his hand

Quite a few people got the answer correct

Very few people are coming tonight because Real Madrid is playing Barca.....  Of course Barca is going to win  :)