While this will go in the Disclaimer thread, abilities were copied from Camp Jupiter Wiki.
Here will be a list of the Gods/Goddess you can have a child of, and also see their powers that come with them.
The Big Three: Chilrdren of the big three can have two of the three sets of powers.
Jupiter (Greek counterpart is Zeus):
Aerokinesis: has absolute control over air.
- can manipulate clouds and any other air structures.
- can generate wind.
- can generate hurricanes and tornadoes, mild, until more kids of Jupiter contribute.
- can manipulate air currents and fly.
Atmokinesis: - control over the weather, mildly. Cannot cause overly drastic changes.
Electrokinesis:
- is immune to lightning.
- can generate bolts of lightning.
- can send static shock through the bodies of others on contact.

Neptune (Greek counterpart is Poseidon):
Hydrokinesis: Control over water.
- can withstand any amount of water pressure.
- can generate water from his/her body, mildly. Not anything like creating an ocean.
- can use sea waves as a form of teleportation.
- can breath underwater.
- can communicate with sea creatures.
Geokinesis: Ability to cause mild quakes.

Pluto (Greek counterpart is Hades):
Necromancy:
- can summon and control the dead.
- can control to an extent the dead and spirits native to the Underworld.
Umbrakinesis: control over darkness and shadows.
- can control darkness and shadows thus using them for concealment as well as attack.
- can travel through shadows (Shadow Travel).
Ferrokinesis: can manipulate and control the riches under the earth's surface.

Minor Gods/Goddess:

Minerva (Greek counterpart is Athena):
- Experts in war strategy
- Wise
- Excellent at crafts and weaving
- Vast knowledge in medicine

Mars (Greek counterpart is Ares):
- Experts in all aspects of warfare
- Extraordinary swordsman and hand to hand combat
- Brave in battle
Telumkinesis: great amount of control over weapons.
- can conjure and use any weapon. Knows everything about a weapon when he sees it.
- amazing physical strength
- can curse weapons for a limited amount of time
- can disarm his opponents.
Odikinesis: has control over feelings and emotions of war (such as hate and rage), and can easily start fights.

Venus (Greek counterpart is Aphrodite):
Beauty and Omnipotent allure: can change his/her appearance at will, depending on the perception of beauty of the person he/she is in the presence of.
Charmspeak: His/her voice has a mesmerizing effect, capable of influencing the emotions of others or of placing them in her/his thrall.
French: speaks and understands French perfectly, as it is the language of love.
Infallible visual acuity: He/she is able to see microscopic flaws in his/her makeup that others normally wouldn't see. Can see it in others too.

Apollo:
- expert archers
- excel in medicines
- skilled musicians and craftsman
- skilled in poetry

Vesta (Greek counterpart is Hestia):
Pyrokinesis: control over fire
- can show others past events
- can summon delicious food

Vulcan (Greek counterpart is Hephaestus):
- prowess in battle
- extraordinary builders
- excels at fighting with a heavy hammer/heavy weapons.
Pyrokinesis: (This is rare) control over heat, fire, and lava. This is a very dangerous and powerful ability, which is why only children that Vulcan chooses have this ability.
- is immune to fire and heat

Mercury (Greek counterpart is Hermes):
- excellent at bartering/being persuasive
- extraordinary at being stealthy and at lock-picking
- very high agility and endurance
- never get motion sickness, and it is unknown of for a child of Mercury to be afraid of heights
- master thieves

Ceres (Greek counterpart is Demeter):
Chlorokinesis: control over plants. He/She can also grant fertility to the earth, allowing plants to grow where they could not grow before. Also, he/she aids plants in growing simply by being near them.
- can transform things into plants, there are limits. You cannot transform a living thing into a plant
- naturally born green-thumbs
- intelligent with different kinds of plants/herbs, making them amazing Medics.

Bacchus (Greek counterpart is Dionysus):
- born party animals, though not all fit in this stereotype
- parties are almost if not always a hit
- can inflict lunacy for a limited amount of time, is stronger with more kids
- cannot become intoxicated if they do not wish to
- ability to control grape vines and use them as weapons
- good actors/actresses, again there are kids who don't always fall into this

Bellona (There is no Greek counterpart):
- fierce warriors, excellent at hand-to-hand combat and armed.
- expert in warfare
Telumkinesis: control over weapons.
- can conjure any weapon. Knows everything about a weapon when they sees one.
- can curse weapons for a limited time
- can disarm opponents.

Fortuna (Greek counterpart is Tyche):
- luck is pretty much always in their favor
- can make others lucky for a limited amount of time, stronger with more kids in it
- can make others unlucky for a limited amount of time

Juventas (Greek counterpart is Hebe):
- have a youthful appearance forever
- usually very handsome/beautiful, never gets pimples, born was perfect teeth, high metabolism rate as well.

Invidia (Greek counterpart is Nemesis):
- has strong sense of revenge and justice
- can be very strategic, especially when it benefits them

Somnus (Greek counterpart is Hypnos):
- can sprout wings for a long duration of time, but must sleep before being able to do so again
Hypnokinesis: authority over sleep. Allows him/her to induce sleep, visions and even manipulate dreams, to an extent.
- can shapeshift into small creatures, biggest would be a medium sized bird.
- can levitate for a medium duration of time

Aesculapius (Greek counterpart is Asclepius):
- will never fall sick
- have the best knowledge when it comes to medicine, making good medics
- healing touch, with one child, they can heal cuts and stop bleeding, but cannot fix broken legs. More serious injuries require more children helping.

Aquilon (Greek counterpart is Boreas):
Aerokinesis: control of air.
- can manipulate air currents
- can generate smaller winds compared to Jupiter kids and control wind.
Cryokinesis: authority over the cold, ice, and snow.
- can freeze things.
- can project jets of swirling ice and snow from his hands.
- can make icy winds blow something out of the air.
- can turn his or herself into anything associated with snow.
- is fluent in French since Aquilon is based in Quebec.

Victoria (Greek counterpart is Nike):
- can influence competition, to a certain extent
- extremely fit
- can influence victory, to a certain extent