How does heat travel from one object to another?
This depends on the conditions of the environment in which the objects are found. For example, the objects can be physically touching each other, they can be separated by a medium (a liquid or gas), or they can be separated by vacuum. Heat will be transferred differently in all three conditions.
In the first two examples, heat is transferred through massive particles interacting with each other. Heat itself is just another way of talking about how much kinetic energy a molecule has, and that energy can be transferred to another molecule by making them collide. When the objects are in direct contact, they directly collide with each other, such as an object on a hotplate. If they are separated by a medium, then the medium will also participate in the heat transfer, such as when an ice cube dissolves in water. Another comparison would be striking two billiard balls together, compared to hitting pins with a bowling ball. In each case, the energy is distributed to each of the molecules interacting in the system.
In the third example, there aren't really any molecules available in the medium to transfer energy from one object to another. Instead, the process of radiation involves the release of energetic photons, a form of electromagnetic energy, that can be absorbed by the second object, exciting its electrons and therefore increasing its kinetic energy.
Heat is just a form of energy. Without intervention from some other force, heat will always transfer from objects of higher heat to objects of lower heat according to the 2nd Rule of Thermodynamics.
Let's assume two solid objects. Two solid objects in direct contact with one another will transfer heat energy by thermal conduction. Although these objects are solid, assuming they have any heat energy (that is, their temperature is not absolute zero, or zero Kelvin), their molecules will still move at some rate. Heat transfer occurs at the molecular level. The objects' molecules move around at a rate proportional to their temperature; when one molecule from the hotter object collides with a molecule from the colder, some of its energy is transferred to the cooler object. Imagine this occurring many, many times between two objects of different temperature and you get some idea of how heat is transferred between objects in direct contact.
If the objects are not touching, more complex heat transfer mechanics may come into play, such as convection, which is heat transfer within a moving fluid, such as water or air. The 2nd Law of Thermodynamics still applies, and heat still is transferred from molecule to molecule, but turbulence can make this process more complex to predict.
Another form of heat transfer between two objects that are not touching is radiation. Any object with a temperature above absolute zero emits thermal (electromagnetic) radiation. This heat radiation can be seen in the infrared (non-visible) spectrum and will also heat up an object over time, in the case of the sun and planet Earth or coals on a stove and meat or vegetables on the stovetop.
Heat transfer also depends upon the specific heat of an object. A material's specific heat determines the rate at which heat transfers to and from a given material. For example, metals generally have a lower specific heat than water or wood, which is why they can feel uncomfortably hot to the touch on a summer day--heat transfers more quickly both into and out of metals than wood. Heat transfers more quickly to your skin from metal than from wood, which may feel more comfortable to the touch even though it is the same temperature as the metal.
Hope this answers your question!