The word dig means to make an opening, usually a hole or tunnel, by removing material. It often refers to removing dirt.
Ex: You need big machinery to help dig a hole big enough for a pool. Something with a blade that will go deep to turn over the dirt before removing it
Ex: I always enjoy digging holes in the sand at the beach. If you get about a foot down, you can find sand crabs.
The word dig can also mean to find something by searching. The common phrase dig in your heels means to remain firm or determined.
Ex: If you dig through those folders on my desk you will find the assignment that you are missing. I do not want you to copy it, but you can read the directions.
Ex: If I dig my heels in now, I will be able to finish this project by morning. They will think I am a genius if my presentation is done this early!
The word bury means to cover something, usually with dirt or water, in order to hide it. It also refers to covering a dead body.
Ex: Pirates are well known for burying treasure. Many movies portray them with shovels, digging the holes that will hold their jewels.
Ex: My mother and father would like to be buried next to each other. I prefer not to think about it and enjoy our time together as a family.
The popular phrase bury the hatchet means to make amends or forgive someone when they have harmed you before.
Ex: I hope that Jennifer and I can bury the hatchet. Apologising is so awkward that we have not been able to sit and talk.
Ex: When I argue with my brother we are always able to bury the hatchet quickly. I usually take the blame willingly so that we can get along again.