Sensible heat is the heat required to change the temperature of matter without it changing state.
Latent heat is the heat required to change the state of matter without the change in temperature. For example going from liquid to solid, liquid to gas, and vice versa.
If you are taking the heat from something, you are cooling that thing and if you are adding heat to something, you are heating it.
Your question about air conditioning is a bit more complex depending on how you look at it.
The air is cool, so it takes heat away from your body. The body temperature changes and that is sensible heat.
If sweat is evaporating, the sweat evaporating is latent heat. However, it absorbs that heat from your body, so the latent heat of the sweat becomes sensible heat for your body.
On the AC side of things. The air flows through an evaporator which as the name implies evaporates the refrigerant inside it. That is latent heat being added to the refrigerant. However, that heat has to come from somewhere, in this case air. The heat coming from the air is sensible heat for the air as the temperature of the air drops. If there is moisture in the air that condenses, that is latent heat instead.
As far as your body is concerned, what you are feeling is sensible heat. If you look at how the heat gets transferred from body -> air -> AC, well it gets pretty intricate as you can see.
Latent heat for any substance is usually given as a fusion or evaporation enthalpy in energy/mass, i.e. J/kg.
Specific heat is what you use to calculate sensible heat. That is given in energy/mass/temperature, i.e. J/kg K. In essence how much heat you need to add to increase a given mass of the substance by one degree.
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