If your little sister is crying because she's scared of the dark and afraid to go to sleep, you could soothe her by singing her a lullaby, or by showing her that there's nothing scary in her room.
Or if your little brother has had a really rough day at school, you could soothe him by hugging him and telling him tomorrow will be better.
Usually we soothe people, but we can also soothe people's pain, or their stress or discomfort. If you suffer a sunburn, you could soothe your burn with an ice pack.
Even if you're not hurt, or not upset about anything, soothing things can still make you feel good. They make you feel calm and peaceful.
For example, soothing music is soft; it can make you feel chilled out or even sleepy. The familiar sound of your father whistling can be soothing to you, and so can the steady roaring of the ocean.