chuckle: laugh quietly, because you are thinking about something funny.
What are you chuckling about? I don't see anything funny.

giggle: laugh quietly and repeatedly because you are nervous or embarrassed.
She spilt the wine on the tablecloth and then giggled nervously at me.

give somebody the giggles: make somebody start giggling.

titter: laugh quietly, unkindly at something embarrassing.
After she ended the lecture, she heard someone tittering.

snigger (UK) / snicker (US): laugh quietly, unkindly at something that is not supposed to be funny.
When the teacher tripped on the steps, the boys sniggered.

roar / howl / shriek with laughter: laugh very loudly.
It was such a good comedy that when it finished, he was still roaring with laughter.

chortle: give a loud chuckle of pleasure or amusement.
When I told her the joke, she started to chortle with delight.

cackle: laugh loudly in a high voice.
When I told her the joke, she started cackling and couldn't stop!

guffaw: laugh noisily.
They guffawed at what their baby had done.

jeer: laugh at somebody or shout unkind things at them.
The president was jeered by a crowd of protesters.

burst into laughter: suddenly start laughing.
The class burst into laughter.

laugh your head off: (informal) laugh a lot and loudly.
He told us a joke after another, and we laughed our heads off!

smile: make one's mouth curve upwards, in order to be friendly or because one is happy.
As the boy left, he smiled at his mother and waved.

beam: smile very happily.
The mother looked at her son and beamed proudly.

grin: smile widely.
When she knew she had won the prize, she grinned broadly.

simper: smile in a silly and annoying way.
He simpered at the boys as he spoke.

smirk: smile in an unpleasant way, to show that you are pleased by somebody's bad luck.
The kids smirked when the teacher fell on the floor.

sneer: smile in an unkind way, showing no respect for somebody.
She sneered at his boyfriend's musical tastes.

laugh in somebody's face: to behave in a way that shows no respect for others.
I told her my opinion and she just laughed in my face.

be laughing all the way to the bank: make a lot of money without making much effort.

somebody will be laughing on the other side of your face: used to mean that although you are happy now, you will be in trouble later.

be laughed out of court (US): be rejected because people think it is completely stupid.
His idea was laughed out of court.

laugh up your sleeve: be secretly happy because you played a trick on somebody or you criticized somebody without him knowing.

laugh something off: pretend that something is less serious than it really is by laughing about it.
The president laughed off rumors that he would resign.

laugh like a drain: laugh heartily.
John liked my joke so much, he laughed like a drain.