Come over:
This command is asking a person to come to you.
I usually use it to ask someone "Do you want to come over?" meaning "Do you want to come over [here] to my house?"
You can also say "come over here" and that means that you want the person to go to where you are. It is the same as saying "come here"
Bring over:
When you ask someone to "bring over," you are asking them to bring something over. So the person should take an object with them.
For example: "Can you bring over a dessert?" means "Can you come here, and have a dessert with you?"
Do not think of "over" being like over/under here. See definition 2.
o·verˈōvər
preposition
1.extending directly upward from.
"I saw flames over Berlin"
synonyms: above, on top of, higher (up) than, atop, covering
"there will be clouds over most of the state"
2.expressing passage or trajectory across.
"she trudged over the lawn"
synonyms: across, around, throughout
"he walked over the grass"