What does the difference between GPP - NPP represent?
The answer is in your overview: it's plant respiration.
Is the value of 800 equal to respiration?
That is the unavoidable consequence of the equation you presented and the numbers you plug in, yes.
I understand that GPP is the overall rate of biomass production by producers at the rate at which solar energy is captured as carbohydrate molecules during photosynthesis and NPP (energy captured per unit error per unit time).
That sentence was fine until "and NPP". What is it doing there? Did you misquote something or are you thinking NPP is a process like photosynthesis?
Producers such as plants use this energy respiration and growth.
I also understand NPP is the rate of energy lost to metabolism or maintenance or the energy stored as biomass by plants or primary producers in the ecosystem.
No. NPP is the energy stored as biomass by plants or primary producers in the ecosystem (or rather the rate at which such energy is stored, if it's productivity you're talking about and it's in gC/m-2/yr-1). Respiration is the energy lost to metabolism or maintenance. GPP is what that sentence seems to be describing, as it's the total carbon captured by plants, i.e. whether it's used to build up the plant's biomass or used by the plant in its metabolism (i.e. respired).
I am making the assumption that this value of 800 is associated with the respiration of plants;
although, I am still confused with what this value represents.
Maybe you should study the metabolic processes of plants and life in general to better understand this. All life consists of chemical reactions that build up structures; in order to build them up you need energy (because of the second law of thermodynamics), and all living things create that energy by breaking down complex molecules into simpler ones. (as such it would be more accurate to say that all life consists of chemical reactions that build up and break down various structures). You might be wondering "but what about the difference between autotrophs and heterotrophs I heard about"; the difference between those is where they get the complex molecules from in the first place. Autotrophs use a different source of energy to build them up while heterotrophs get them from their environment. As such, you can think of every living thing as being made of two kind of molecules: those that actually form their structure (in humans, the molecules that make up cell membranes, bones, muscles, etc) and those that are stored in order to be broken down to power the whole system (in humans that's fat, glycogen, glucose, etc). Of course a molecule can do both; if you're starving your body may start to break down structural molecules for power. There are many different ways of breaking down those big molecules for power; the most efficient one, that starts with a big chain of carbon atoms and cuts it down into individual CO2 molecules using O2 molecules, is called aerobic respiration (i.e. respiration that uses oxygen).
Because those complex molecules are required to power all life, autotrophs (the organisms that actually make them) are very important, and the processes they use to make them are very important too. The process that makes almost all of the molecules that power almost all life on earth is photosynthesis, which uses the energy from the sun to power a reaction that converts CO2 from the atmosphere into big carbon-based molecules we'll call carbohydrates. This is called "fixing carbon", since the carbon atom is the most important one; measuring how much photosynthesis is happening is another way of measuring how many carbon atoms move from being part of a CO2 molecule to being part of a plant.
GPP is the measure of that exact process: how many grams of carbon per square meter per second (or year in your graph) move from being in a CO2 molecule to being part of a plant via the process of photosynthesis.
You might think this is also a measure of how much the plant grows, but it ignores the fact that as I said, all living things contain molecules that make them up and molecules they break down for power; obviously the plant's growth will involve the former, and the latter just goes back to being CO2 through the process of respiration. Another way of putting it is that in the flow of carbon, there is a huge flow of CO2 to plant molecules via photosynthesis (GPP), but a huge proportion of that flow goes right back the other way (the plant molecules are broken back down into CO2 via the plant's respiration); the remaining flow that's actually still in plant molecules at any given time is NPP.
NPP is the proxy for plant growth; it's basically how much of the carbon the plant fixed via photosynthesis it actually contains in its body, while R (respiration) is how much of the carbon it fixed it broke back down for energy, releasing it as CO2 into the atmosphere. Hence, GPP = NPP + R, and GPP - NPP = R.
From my way of thinking, the difference between GPP-NPP equals the extent of CO2 lost from plant respiration within the ecosystem.
Since the value of GPP (1400) is larger than the value for NPP (600), more energy has been used during the intake of carbon during photosynthesis rather than plants using glucose or the energy stored in the system to photosynthesise for the emission of CO2.
I don't understand this sentence at all. The fact that GPP is larger than NPP shows that plants don't keep all of the carbon they convert from the atmosphere to make up their physical structure; they also use a lot of this carbon to power their metabolism through the process of respiration, which results in said carbon turning back into CO2 and being released back into the atmosphere (as opposed to staying in the form of wood, leaves, starch, etc).
Therefore, this value represents the amount of CO2 that is lost from plant respiration or from metabolic activity.
That is correct, with the two caveats that it's the amount of carbon fixed in organic molecules that is lost by being released into the atmosphere under the form of CO2, and that there is no "or" - all plant metabolism is fuelled by respiration, and the only metabolic activity that releases CO2 is respiration.