Many people use a microwave oven to cook their food. When it is turned on, what type of energy transfer is taking place?

I will discuss forms of energy first (and how they transform), and then it will be easy to identify which means of energy transfer occurs (conduction, convection, or radiation).
The name microwave oven gives us a clue to the form of energy it produces: microwaves are electromagnetic waves on the spectrum between radio waves (longer wavelength and lower frequency) and infrared waves (shorter wavelength and higher frequency than microwaves). All electromagnetic waves, including microwaves, are a forms of energy that may be called electromagnetic energy or radiant energy.
What is the energy input? We plug a microwave into the wall; the wall current is sometimes simply called electrical energy. We can be more specific and note that the quantity voltage is also called "electrical potential"—this is the potential energy that electrons have relative to the ground. So you could also call the energy supplied to the microwave potential energy.
In the process of cooking food, the radiant energy of the microwaves interacts with water molecules in the food, causing them to vibrate (or exciting molecular vibrations). This is vibrational energy. It is quickly transferred between molecules in the food, causing more kinds of molecular motion. In solids, most of the motion is vibration. In liquids, the molecules can move (translation) and tumble (rotation). Taken together, these molecular motions are called thermal energy in physics but is often referred to as "heat" in chemistry.
So the energy in a microwave starts out as electrical potential energy, is transferred to the food as radiant energy, and ends up as thermal energy or heat. Because the transfer from the oven to the food occurs by means of microwaves—which are a form of radiation—this transfer occurs by radiation. It is not conduction, which requires objects to be in contact, nor convection, which involves movement within fluids.