With is used in a number of expressions that say how people express their emotions and sensations. Common expressions are: green with envy, blue with cold, white with fear or rage, red with anger or embarrassment etc.

The old woman was shivering with cold.
She was trembling with rage.
The boy was jumping up and down with excitement.

With means against when used after words like fight, struggle, quarrel, argue and play.

Don’t play with fire.
Will you play with me?

With is used after some adjectives which say how people are feeling towards others.

Common expressions are:
patient with
angry with
pleased with
happy with
upset with
furious with

Note that with is not used after words like nice, kind, polite, rude and good.

She was very nice to me. (NOT She was very nice with me.)

He is rude to everybody. (NOT He is rude with everybody.)

With is often used to indicate possession and similar ideas. Note that its meaning is similar to have.

I need a computer with a fast processor. (= I need a computer which has a fast processor.)
We are looking for a house with a garden. (= We are looking for a house which has a garden.)

Note that when referring to articles of clothing, we usually use in instead of with.

Who is that boy in the red shirt? (NOT Who is that boy with the red shirt?)