burst into tears: suddenly start crying.
When her boyfriend told her the truth, she burst into tears and ran out.

break down in tears: suddenly cry a lot, after trying not to cry.
After reading his letter, she broke down in tears.

be close to tears / be on the verge of tears: be about to cry.
When she heard his voice on the phone, she was close to tears.

have tears in one's eyes: be about to cry.
When I said goodbye, I had tears in my eyes.

shed tears: cry.
I must admit I shed a few tears when the school closed.

be in tears: be crying.
The children were all in tears when our dog disappeared.

be moved to tears: be so upset that you start to cry.
A lot of people were moved to tears by his story.

weep: cry a lot for a long time.
The kids wept bitterly when it was time to leave.

cry one's eyes/heart out: be extremely sad and cry a lot.
After the robbery, she cried her heart out.

bawl: (a baby) cry very loudly.
We could hear the baby bawling upstairs.

sob: cry noisily, with sudden noisy breaths.
He began sobbing uncontrollably.

your eyes water: you start to cry, especially because there is a lot of smoke or because you have been cutting up onions.
There was so much smoke in that room that my eyes were watering.