You swear to never mention what you've seen. Swear by my sword...
Horatio (Act 3, Scene 2, Page 2)
Here, sweet lord, at your service.
Horatio, you're the best man I've ever known.
The scene between Bernardo, Marcellus, and Horatio is important. Despite Marcellus having seen it twice, Horatio doesn't believe there is a ghost. This will show us the type of character Horatio is later in the story. This is because Hamlet believes in the ghost and Horatio still helps him and looks out for him.
This shows significance because Horatio finally believes in the ghost. Hamlet asks him to swear by his sword not to tell anybody about it. Horatio does, like the good friend he is, and this shows character and loyalty to Hamlet as his only true friend.
This shows significance because Hamlet has dialed his list of friends to one person. He believes Horatio is his only friend. Horatio shows his loyalty by telling him he is there at his service, ready to help him. Hamlet and Horatio truly are best friends, and this scene proves it.