There are many spells and rituals that fall under this category. You may want to banish negative thought from you, banish unwanted energy around you, or even banish an unwanted person from your life. Here are some I have tried and some I haven't.

A Ritual for the Dark Moon
Begin by setting up your altar as you normally do for ritual, making sure you have a black candle and wine or water in your chalice in addition to anything else you use. You will also need a cleansing, purifying incense and a censer or some kind of container which can be carried around your home. It is best to set your altar on the floor for this rite, but you may do it however you feel most comfortable.

Cast your circle as you normally do. When you are through, kneel before your altar and light the black candle. With your arms held, palms upward, atyour sides, call Hecate by saying three times:
"Hecate, beautiful Crone of Night
I call you here to put things right.
Transform the negative thought and pain
And help my life be whole again."

Close your eyes. When you feel the presence of Hecate and know she is there to help you, open your eyes. Bow your head to her to show your reverence, and then take the chalice, saying:

"Lady of the Dark Moon, Share with me this wine.
Bring your protection to Flood this life of mine. May the waters of your eternal womb
Bring change most divine."

Sip a small amount of the wine (or water), envisioning it as liquid energy, flowing to effect a positive change within and outside of you. Leave the rest as an offering to Hecate. Light the purifying incense in the censer, cut a door in the Circle, and, beginning in the eastern-most corner of your home, smudge your home, going clockwise. Go into every closet, the bathroom, and the garage - make sure your entire house is smudged. As you go, chant:
"Negativity be gone."

Come back into the Circle and visualize your entire house and yard bathed in a peaceful blue light. Since you have created a void by banishing the negativity, you will need to fill that void. Ask that good, protective spirits come into your home and that positive energies replace the negative ones that have just been banished. This is an important step, because if you don't fill the void with something good, the negativity will come right back.

Sometimes during such a ritual, I take one large and one small black stone and charge them to keep away negativity. I promise Hecate that the large one will remain in a prominent place in my house and that I will keep the small one with me at all times. Thank Hecate. Meditate if you wish, visualizing your life free from negative happenings and feelings and full of love, prosperity, and happiness. Feel how She has changed your home and your life. Close the Circle and know that it is done.

Separating Yourself from a Problem or Person
This is to Separate yourself from a problem or a person. Small issues can be handled in a magick circle, whereas larger issues may need a full ritual. Think carefully which is best for the situation. You will need two black candles, a long black ribbon, black gloves, scissors, the cauldron containing small amount of alcohol, and a silver bell. Cleanse and consecrate all items. Dress the candles to send negativity away from you, then light them. Hold each end of the ribbon and name the ends. Pass the silver bell over the altar several times, until you feel you are in tune with the Universal energies of love and peace. Anger is not a luxury afforded to you in the situation. Put on the black gloves and say:

I call upon the energies of Universal balance
I call upon the ancient energies of my people
I call upon the living essence of the Morrigan
Underworld and heaven
Land and sea
I waken these energies unto me.
Witness now that I renounce and sever
Break bonds and connections with ______
By all powers that are One Power
May the great sisters of karma now weave anew
And separate me from you
As I will, so mote it be!

Cut the ribbon:

It is so.

Light the cauldron. Burn both pieces of ribbon.

Simple Banishing Ritual

This is just a simple banishing spell. It's for when you continue to let the past run your future and the like.

Materials: a white piece of paper, no lines, any size is fine, A writing utensil, a heat proof container to put the papaer in and A lighter

Write the name of the person who is constantly in your thoughts three times on the sheet of paper followed by an affirmation, such as "I will no longer allow you to have control over my life." Read the affirmation out loud to yourself.

While continuing the affirmation, burn the piece of paper.

See also Correspondences for Banishing Spells and Banishing Spells Grouped by Situation.