After the discussion of the Sectumsempra spell, I made this list of spells, their incantations and, most importantly, how they work based on what we see in canon.
Incantation
Spell name
Spell description including limits and any common RP mistakes I see when people use it.
Canon Spells
Accio
Summoning charm
This spell retrieves an object for the caster, however the person casting the spell must be familiar with the object and be aware of its approximate location. Objects cannot be summoned over infinite distances.
Agumenti
Water charm
This spell causes a jet of water to shoot from the caster's wand.
Alohomora
Locking charm.
This spell locks and unlocks, well, locks. Doors which have been magically warded or sealed cannot be opened with this charm.
Anapneo
Healing charm
This spell clears the airway of the patient, if blocked.
Anti Disapparition Jinx (incantation not given)
This spell prevents a person from apparating.
Antler Jinx (incantation not given)
This spell makes the person it hits spontaneously sprout antlers.
Aparecium
Visiblity spell
This spell causes invisible ink to become visible.
Avada Kedavra
The killing curse.
This spell creates a blast of green light. It cannot be cast non-verbally. It cannot be blocked.
Avis
Bird Conjuration
This spell conjures a flock of birds.
Babbling Curse (incantation not given)
This spell causes whoever it hits to babble uncontrollably.
Banishing Charm (incantation not given)
This is the opposite of a summoning charm. It sends the object back to wherever it is normally kept.
Bedazzling Hex (incantation not given)
This spell appears to cause an object to become dazzling, thereby making it invisible to anyone who looks directly at it.
Bat-Bogey Hex (incantation not given)
This spell engorges the victim's snot to the size of bats and sets them to attacking his or her face.
Bubble-Head Charm (incantation not given)
This spell creates a bubble of air that is safe to breathe for up to one hour.
Caterwauling Charm (incantation not given)
This spell creates a wailing noise. It can also be used as an alarm.
Cave Inimicum
Protection Spell
This is spell stops enemies from entering an area.
Cheering Charm (incantation not given)
This spell cheers people up.
Colloportus
Sealing spell
This spell creates a magical seal. It cannot be undone by Alohomora.
Confrigo
Blasting Curse
This seal causes the object it hits to explode.
Confundo
Confundus Charm
This spell causes the person it hits to become confused. It is not a memory modifying charm.
Conjunctivitis Curse (incantation not given)
Affects the vision of the victim.
Crucio
The Crucatius Curse.
This spell causes a flash of green light. It cannot be cast non-verbally. It cannot be blocked. It causes unbearable pain. Prolonged exposure to it causes mental illness.
It is commonly assumed that the cruciatus only sets off the pain receptors in the brain, however as the word crucio literally means torture, it is possible that there are subtle physical manifestations as well.
Deletrius
Deleting spell
This spell deletes the spell traces left by Priori Incantatum. Possibly deletes other spell effects as well.
Defodio
Gouging spell
This spell gouges a hole in something.
Densaugeo
Tooth engorgement charm
This spell causes the victim's teeth to become engorged.
Deprimo
Blasting Charm
This spell blasts a hole in something. I assume this is just a more forceful version of defodio.
Descendo
Lowering Charm
This spell causes the object it hits to descend.
Diffindo
Severing Charm
This spell causes cuts. This is the non-dark cutting spell that can be healed. Use it instead of Sectumsempra.
Dissendium
passage opening charm
Opens the secret passage in the statue of the humpbacked witch on the third floor of Hogwarts. The passage leads to the Honeydukes storeroom in Hogsmeade.
Duro
Petrification charm
This spell causes the target to turn to stone.
Engorgio
Engorgement charm
This spell causes the object to swell in size
Entrail Expelling Curse (no incantation given)
This spell causes the victim's innards to come out of them.
Episeky
Healing Charm
Heals minor injuries. Serious injuries need to be seen to by an experienced healer.
Erecto
Straightening Charm
Straightens the object out.
Evanesco
Vanishing Spell
This spell causes the target to cease existing.
Expecto Patronum
Patronus Charm
Creates an incarnation of the good feelings of the caster, either in the form of an animal or a silvery emission from their wand that may protect them from dark magics. The patronus is the wizard or witch's spiritual assistant. They can also be used to carry messages.
Expelliarmus
Disarming Charm
Strips the victim of his or her wand. Does not propel the wand into the hand of the caster.
Expulso
Exploding spell
This spell causes the target to violently explode.
Ferula
Rod spell
Conjures a wooden rod
Fiendfyre (incantation not given)
This spell causes a semi-sentient fire which seeks out life to burn. The forms of dangerous beasts can be seen in its flames. It can only be put out by an experienced wizard or witch who knows the counter spell.
This spell should be used rarely and only by upper level dark witches and wizards. To start an ordinary fire, use Incendio instead.
Finite
Ceasing spell
Stops one effect of a spell.
Finite Incantatum
Ceasing spell
Stops all the effects of a spell.
Flagrate
Flagrate Curse
This spell causes the object it is cast on to burn anyone who touches it.
Furnunculus
Boil Curse
This spell causes painful boils to break out all over the victim's body.
Geminio
Duplication spell
This spell creates duplicates of whatever it is cast on.
Glisseo
Smoothing spell
This spell converts an uneven surface into a smooth one.
Homenum Revelio
Human revelation spell
Reveals the presence of humans in the area.
Impedimentia
Impediment Jinx
This spell stops or slows the thing it's cast on down.
Imperio
Imperius Curse
This spell has no visual effects. It can be "thrown off", but the ability to do so is rare. It can not be blocked. It places the victim under complete control of the caster and they will carry out the caster's whims even at a distance.
Impervius
Impervius charm
This spell makes the thing it is cast on water-repellent.
Incarcerous
Binding spell
This spell binds the target with thick ropes.
Incendio
Fire starter charm
This spell starts a fire. This is the spell your character should be using instead of fiendfyre.
Insect Jinx (no incantation given)
This spell forces the victim to scuttle around on the floor like a cockroach.
Jelly Brain Jinx (no incantation given)
This spell affects your ability to think
Jelly Fingers Curse (no incantation given)
This spell makes your fingers all wobbly and makes it hard to hold things
Jelly Legs Jinx (no incantation given)
This spell makes your legs all wobbly like jelly
Knee reversing hex (no incantation given)
this spell reverses the victim's knees, making it difficult for them to walk
Langlock
muting charm
This spell causes the victim's tongue to adhere to the roof of their mouth.
Leek ear jinx (no incantation given)
This spell makes leeks sprout from the ears of the victim
Legilimens
Legilimency
If the caster is skilled in legilimency, this spell allows them to access memories and feelings of their victim. It does not allow them to read the victim's mind.
Levicorpus
levitation charm
This spell catches the victim by their ankle and hoists them into the air, leaving them dangling.
Liberacorpus
descending spell
This is the counter spell to Levicorpus.
Locomotor
locomotion spell
This spell causes an object to rise into the air and move where the caster commands it. It is usually used in conjunction with the name of the target object ("locomotor trunk")
Locomotor Mortis
leg locker curse
This spell causes the victim's legs to lock together, making them unable to walk.
Lumos
light charm
This spell shines a beam of light from the tip of the caster's wand.
Lumos Solem
This spell appears only in the films. It is not canon. Additionally, devils snare shrinks from fire, not sunlight.
Meteolojinx Recanto
meterological spell
This spell stops it from raining.
Mobilicorpus
locomotion charm
This spell moves a body.
Morsmorde
The Dark Mark
This spell conjures the dark mark.
Muffliato
tinnitus charm
This spell fills the victim's ears with a ringing noise.
Nox
darkening charm
The counter charm to lumos. This charm turns out enchanted lighting.
Obliviate
Memory modification charm
Modifies a person's memory. If carelessly performed, it may erase a person's memory completely.
Obscuro
blindness charm
This spell obscures a person's vision.
Oppugno
attacking charm
This spell causes conjured creatures to attack.
Orchideous
flower charm
Conjures a bunch of flowers
Pack
packing charm
Packs your bags for you.
Petrificus Totalus
Body-Binding Curse
This spell turns the entire body of the victim rigid.
Piertotum Locomotor
animation charm
A variation of the locomotor charm, it animates statues and armour within Hogwarts.
Point Me
Four Point Spell
This spell causes the caster's wand to point north.
Portus
Portkey charm
This spell creates a portkey.
Prior Incantato (singular; priori incantatum is plural)
previous incantation
Causes a wand to produce a ghost effect of the last spell it cast.
Protego
Shield Charm
Creates a barrier that will deflect minor spells.
Protego Horribilis
Shield Charm
Creates a barrier that will deflect dark spells. Will not protect against the Unfogivables.
Protego Totalum
Shield Charm
Creates a barrier that will protect an entire area. Used on buildings and places.
Quietus
quieting charm
Makes the sound of a person's voice quieter. Acts as the counter-spell to sonorus.
Reducio
reduction charm
This charm returns an engorged object to its normal size.
Reducto
Reductor Curse
Blasts solid objects out of the caster's way. Does not always destroy the objects.
Relashio
fire spell
releases a jet of fiery sparks
Rennervate
reviving charm
Used to revive someone who has been hit by a stunner.
Reparo
repairing spell
Used to repair objects.
Repello
repelling charm
Keeps an object away from the caster.
Repello Muggletum
muggle repelling charm
Repels muggles from an area
Rictusempra
Tickling charm
causes a tickling sensation
Riddikulus
Boggart banishment
Along with imagining your worst fear in a silly situation, this spell is used to banish a boggart.
Scourgify
scouring spell
cleans things
Sectumsempra
slashing spell
This spell creates a deep wound or wounds that must be healed by a healer. It should not be RPed as an ordinary cut or wound.
Serpensortia
serpent conjuration
Conjures a snake
Silencio
Silencing Charm
Silences something.
Sonorus
volume charm
Makes the caster's voice louder
Specialis Revelio
Scarpin's revealaspell
Reveals the components in a potion or enchantments on an object.
Stupefy
Stunner
Causes the victim to become unconscious
Tarantallegra
tap dance jinx
forces the victim's legs to do a crazy dance
Tergeo
cleaning spell
tidies up an object or person
Waddiwasi
unsticking charm
Causes something that is stopped up to become unstuck.
(wingardium) Leviosa
Levitation charm
The incantation suggests that wingardium leviosa is the spell specifically for levitating feathers. It is likely that Leviosa by itself is the incantation for levitation.