Flames and flame Strengths

Flame: Sky

Color: Orange

Characteristic: Harmony (Armonia)

Purpose: "The Sky; one that colors and engulfs everything"

Description: Those who possess Sky Flames are exceedingly rare, and possibly the most powerful people in the world. The characteristic of harmony is said to maintain the balance between the other Flames, and contains neither contradiction nor conflict.

Appearance: The Sky Flame seems to have an Orange center for the Flame, then a pale Orange for the outer spiking, that of which is similar in appearance to Rain Flames, with largish oscillating spikes.

Flame:Storm

Color: Red

Characteristic: Degeneration (Distruzione)

Purpose: "The Raging Storm; one that fiercely blows away everything"

Description: Anything that touches Storm Flames begins to break down and corrode, even other Flames. These Flames are highly offensive.

Appearance: The Storm Flame appears to be almost opaque in appearance. With a Red Color at the center, then an inner area of a very pale red-white, and an outer spiking of deep red coloring resembling a real fire. In the manga, the ends of the Flames are distinctively dark.

Flame:Rain

Color: Blue

Characteristic: Tranquility (Calma)

Purpose: "The Requiem Rain; one that washes away everything"

Description: These Flames weaken other Flames and Attacks such as Illusions, rendering them almost completely ineffective. The stronger the Flames, the weaker the enemy's Attacks become.

Appearance: The Rain Flame appears to have sparkles of some type running through it, giving it a similar appearance to Water. It is more or less opaque, and has a Dark Blue at the center while the outer part of the Flame is a pale Light Blue with large, oscillating spikes.

Flame:Sun

Color: Yellow

Characteristic: Activation (Attivazione)

Purpose: "The Sun; one that illuminates the Sky"

Description: Sun Flames initiate processes and speed them up at an incredible rate. Difficult to use offensively, the activation can be used effectively for healing and power up purposes.
Appearance: Sun Flames have a constant Yellow coloring throughout the whole Flame, and circular "shines" of light emanating from it as well

Flame: Lightning

Color: Green

Characteristic: Solidification (Indurimento)

Purpose: "The Lightning; one that harshly strikes everything"

Description: These Flames are the most solid flames and have the ability to harden anything infused with the Flames, increasing their offensive and defensive properties. This Flame is very unique as it has more of an electrical quality than actual Flame, as it is used to electrocute opponents rather than burn them.

Appearance: Lightning Flames have the appearance of Lightning. The center is an off-white Green, and the extremely sharp "spikes" of Flame are a Neon Dark Green in color. They have a tendency to form round shapes with the sharper part flailing around, like how electricity is often times shown.

Flame: Cloud

Color: Purple

Characteristic: Propagation (Propagazione)

Purpose: "The floating Cloud; one that cannot be caught and goes its own way"

Description: Cloud Flames and Cloud Flame-based Weapons absorb other Flames to grow in size or multiply, and can even cause phenomenal growth in human beings if their bodies are altered. Hibari can use the Flames like a shield, as been seen on Episode 130, when he defends Genkishi's attack with his Flames.
Appearance: These have a dark Purple at the center, and a Pale Light Purple for the rest of the Flame. And unlike Rain Flames, which have sharp oscillating spikes, Cloud Flames have small rounds of flame instead of points.

Flame: Mist

Color: Indigo

Characteristic: Construction (Costruzione)

Purpose: "The Mist, one that cannot be captured"

Description: Mist Flames are capable of generating Illusions and constructing them into real, solid objects.

Appearance: Mist Flames have an off-white indigo for the majority of the Flame, and a gradually darkening Indigo for the rest of the Flame deepening outward.
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Credits to flame descriptions to Reborn Wikia
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Flame Weakness

Storm>Cloud>Rain>Sun>Lightning>Mist>Storm
Damage increased by 1.5x if attack is flame based and is strong against that flame.
Flame damage against a flame that the flame is weak to is halved.
Sky is neither weak or strong to anything.

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What is Flame? Well it is also known as Dying will Flames. Here is a short explanation.

The Dying Will Flame is the basis for Tsunayoshi Sawada "resurrected" strength in the beginning of the series but over time, becomes a more crucial part to the story. In the Future, all combatants use their own Flame to fight one another. It is highly suggested that different Flames are within one's bloodline; so, a type of Flame could be shared and/or inherited within one's Family.
Flame Waves
Flame Attributes
There are seven different Flames that match the title of Guardians of the Vongola Famiglia: Sky, Storm, Rain, Sun, Cloud, Lightning, and Mist. Each Flame is identified by one of the colors of the rainbow and possesses a unique characteristic. Dying Will Flames are the source of power used to activate the Box Weapons in the Future, and can cover Weapons, like Swords, to strengthen and reinforce them. Reborn explain it's a overly dense version of aura. However, unlike aura, the Dying Will Flame is a real Flame that can melt, burn, and even burn up the oxygen in the air. A Dying Will Flame is super condensed energy, unlike any aura. The purer the Flame is, the more powerful it is, and the greater it demonstrates its characteristic.