How are the following forms of energy significant to organisms in ecosystems?

Radiant is sunshine which helps in photosyntesis
mechanical is for multicellular that are not plants to help them move
chemical is for internal functionality
and heat is to keep the exchange of energy so the organism wont die

This is a very complex question covering the whole of an organism's metabolism. Lets break it down piece by piece, OK?

Radiant energy: An obvious example of this is sunlight, which is the fundamental fuel for our ecosystem at its most basic level. (With the exception of some dee-sea bacteria, and other organisms) Plants, and more basically, cyanobacteria, use sunlight as part of the photosynthetic reaction used to store "food". What is FOOD though?

Chemical Energy: That's all that sugar is, and all that plants are to a cow: stored chemical energy. Now the energy which has been in an ambient form (sunlight, and more) is used to create long strings of hydrocarbons, or sugars, starches, and even proteins. All of these complex molecules have Chemical Binding Energy, and some (sugars for humans) give a high yield when metabolized. The next order from CBE, is Nuclear Binding Energy, the result of which can be seen in nuclear fission and fusion. Beyond that is something called 'Degeneracy Pressure', which is involved in keeping White Dwarfs and other compact stellar objects from total collapse to a Black Hole. I tell you all of this to give you an idea of just how important Chemical Energy is to organisms of all types, because we all need it to DO things like reproduce which takes...

Mechanical Energy: This is easy, and it's just what it sounds like; when you move, that 'expends' mechanical energy (from the point of view of an organism. The energy doesn't go away, but that's next...). ME is divided for the sake of this answer into Potential Energy, and Kinetic Energy: You need the former to get the latter (the first to get the second), and so we eat. Our metabolisms, and those of every mammal on Earth as one example: ultimately break down complex carbohydrate molecules into simple sugars, eventually Glucose. Glucose then incorporated through the various metabolic pathways used by organisms into ADP, ATP, and other simple compounds which release a great deal of Chemical Binding Energy, allowing for a hgh efficiency of transfer from Radiant -> Mechanical. In other words, we 'burn' chemicals in our mitochondria to fuel every cell in our body: that is the basis of metabolism. It's not the only way to metabolize sugars, but it's a decent example. So, what happens when you burn anything, chemically or otherwise? Unless you can be 100% efficient in using your chemical energy... which is impossible... even then Mechanical exertions are also not 100% efficient. When there's is something less than 100% efficiency in a system, you have waste; in the case of metabolism it's mostly chemicals that are excreted. In the case of mechanical energy you get...


HEAT: Heat is... a kind of Radiant Energy. This brings it all home, because we've all moved, and we've all felt our bodies get hotter and radiate that heat. That's energy that our body is wasting; literally "waste-heat", and it becomes part of the ambient energy in the room, mostly in the form of far Infra-Red EM Radiation. Eventually that radiation is absorbed... then re-emitted as:

Radiant Energy.