What is a sleep disturbance? Do you have to wake up or can you sleep through it?

Disrupted Sleep. A "sleep disturbance" is anything that disrupts the normal sleep pattern. So, it could be sleep apnea that keeps waking someone's brain up. Or, frequent urination; or insomnia from a number of reasons. Usually, you don't "sleep through" a sleep disturbance because it would not be a disturbance then.
Sleep disorder. Many sleep disorder, like sleep apnea and periodic limb movements the person may not be aware of. Likewise, waking up a few times at night briefly is not a sleep disorder. What counts is how restful your sleep is in the morning. If you sleep excessively and are still tired that might be a sleep disorder which only a sleep study can identify correctly.
Sleep disturbance. If you are waking up multiple times per night please have an evaluation from a psychologist expert in sleep disorders. Waking up from dreams can definitely be resolved. In my book and class, No More Nightmares, I teach Planned Dream Intervention which teaches the sleeping brain that there is no such thing as a bad dream. You end up sleeping peacefully through any dreams.