When energy is transferred from a warmer object to a colder object, we call that energy heat. Heat moves in a combination of three ways. Conduction is heat transfer by direct contact of the warmer and colder objects. When we touch and object and feel how hot or cold it is, that is conduction. Convection is heat transfer through a fluid medium. When air or water brings heat from a boiler to the rooms of a building, that is convection. Radiation is heat transfer through light waves. When the part of you that faces the firelight gets warm, but the part in your shadow does not, that is radiation.
Conduction is also called heat diffusion and can happen between objects or from one part of an object to another part. If a material conducts heat well, it is a thermal conductor. If is conducts poorly, it is a thermal insulator. Metals are good conductors; we can quickly feel whether thay are warmer or colder than our skin. Wool and Styrofoam^tm are good insulators; they slow down heat conduction, but don't stop it completely.
In convection, fluids mix to carry heat from one place to another. If the source of heat is below the surface of the fluid, then we get convection currents. Colder fluid is usually denser than warmer fluid, so gravity pulls the colder fluid down more and the warmer fluid floats up. Then colder fluid then gets heated up to repeat the cycle until all the fluid reaches the same temperature (or the heat source is removed). This is an important part for explaining ocean currents and various weather phenomena.
Thermal radiation (which is different from nuclear radiation) travels mostly as infrared light waves which are invisible to human eyes.

Through conduction, convection or radiation, heat moves from warmer objects to colder objects. Usually, more than one method is used. Once the two objects (or the two parts of an object) are the same temperature, it reaches thermal equilibrium. Then, no heat flows until another colder or warmer object is introduced, giving heat a place to flow to or from.
All objects radiate out thermal radiation, but if objects are the same temperature, then the radiation out and in to each unit of surface area is the same and thermal equilibrium is reached because the net flow iz zero.

When heat moves into an object, that object's internal energy goes up. This internal energy is the energy of the atoms and molecules inside of an object. If a batch of two trillion iron atoms speed up their motion, the total energy increases and it might begin to glow red, become pliable, or even melt. If a batch of four trillion atoms (double the mass) has the same temperature, then the atoms will be moving at the same average speed, but it will have twice the total energy. Temperature is the average internal energy of the atoms and molecules in an object. If we could actually see the particles moving around in an object, some are moving faster, some slower, but the temperature would stay the same as long as the average motion stays the same.
Temperature can be measured in degrees Fahrenheit, degrees Celsius, or Kelvin. At about 0°F ocean water will freeze. At 32°F pure H₂O will freeze and 180 degrees higher at 212°F pure water will boil. Fahrenheit is a good scale for everyday human experience since we generally experience a range 0-100°F. At 0°C pure H₂O will freze and 100 degrees higher at 100°C it will boil. Celsius is a good scale for science exploration, since substances often need to be a much higher temperature to burn or melt or boil or react with something else. The Kelvin scale is simply 273 plus the Celsius degrees, so pure H₂O freezes/melts at 273 K and boils/condenses at 373 K.
If an icecube is dropped into a cup of hot tea, the icecube will absorb energy from tea and change from a solid (ice) to a liquid (water), while the tea gets cooled down. To get water to boil to a gas takes energy from a stove or microwave or oven. At the surface of a liquid, some faster, more energetic liquid particles can escape into the air, leaving the slower, cooler particles behind. This happens when we sweat and water evaporates from our skin to cool us down. If our sweat cannot evaporate, maybe because the air is very humid or we accidentally wore a sweater to the beach, then our body has a tough time cooling down. Some animals, like dogs, don't sweat through their skin, but evaporate water through their tongues and breath.
Just like we can put energy into something to get it to change, it can change phase the other way if we remove energy. When you see "steam" above a cup of hot drink, you're actually seeing the invisible steam turn back into a cloud of water when it releases energy into the colder air. When an air conditioner cools the air, the humidity (water vapor) in the air gets turned into liquid water which has to drain out. Something similar happens to form dew at the end of a cool night. During a cold morning, the dew releases even more energy to freeze and make frost. This frost can appear as ice crystals on the grass in a yard or the glass of a car. Since the colder particles condense or freeze and the warmer particles don't condense or freeze, the air that carries a storm actually gets a bit warmer when it rains or snows because it drops the slower moving (cooler) particles down to the ground.

When an object changes phase (freezes, melts, boils, evaporates, or condenses), a certain amount of energy is absorbed or released without changing the temperature of the object. When the temperature does go up, it's only because heat is absorbed. When its temperature goes down, then energy must be released. To cool down a room by 20 degrees, twice as much energy needs to be absorbed than cooling the same room by 10 degrees. To heat up two cups of tea takes twice as much energy as heating up one cup by the same amount of degrees. So, the amount of heat we need to change an objects temperature depends on how much we are heating or cooling it and how much stuff we need to heat up or cool down.
The amount of heat also depends on what the material is that we're playing with. 4.2 Joules of energy will heat 1 gram of water by 1 Celsius degree. To get 1 gram of a metal like iron to heat up by 1 degree Celsius only takes about 0.5 Joules. So, 210 Joules of energy would heat 10 grams of metal (about 2 quarters) by about 42 degrees and heat 10 grams of water (about a big sip) by only 5 degrees. The rate at which an amount of substance will change temperature with heat is called specific heat capacity. It is also sometimes called "thermal inertia". I like "hotness resistanceness" or "coldness resistanceness"; hard to say but easy to remember.
The formula relating heat (ΔQ), mass (m), specific heat capacity (c) and temperature change (ΔT) is:
ΔQ = m · c · ΔT
When specific heat is high, that material needs to absorb or release more heat to get hot or cold. If the same amount of two materials get the same rate of heating or cooling, the material with the lower specific heat will get hotter or colder faster. Sometimes the mass and specific heat capacity are multiplied already to give a (non-specific) heat capacity for an object (like a calorimeter). Then, the formula becomes ΔQ = C · ΔT, which is a bit simpler, but no less easy.