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In the lemon battery experiment, as I understand, both $\ce{Zn}$ and $\ce{Cu}$ meet acid in the lemon, cause $\ce{Zn}$ and $\ce{Cu}$ release electrons to form $\ce{Zn^{2+}}$ and $\ce{Cu^{2+}}$.
So why do the electrons follow from $\ce{Zn}$ to $\ce{Cu}$ but not from $\ce{Cu}$ to $\ce{Zn}$?
In fact $\ce{Cu}$ doesn't release electrons, only $\ce{Zn}$ does. For the usual explanation see this Wikipedia article, I will try to explain it in other words to make it much easier.
One important thing that you should know is that energy comes from metal not from the acid.
Some species have the tendency to donate electrons other species the tendency to acquire them. Chemically speaking some species have more internal energy when their are in the reduced form (when they have the maximum number of electrons) others species reach their minimum internal energy when they are in the oxidized form (when they have the least number of electron). In this world every system tend to go to a state of lower energy spontaneously so if two species are not in their lowest energy form if they can exchange their electrons freely they acquire (are reduced) or give (are oxidized) electrons until they reach the lowest energy. There are also species that do not have any particular tendency or are only little more prone to acquire electrons or to donate them. In all these cases the exchange can occur only if the electrons can move easily from one specie to another, for this reason a good electrolyte is many times needed.
We could imagine chemical species as men and electrons as money. Imagine that there are different kind of men at the antipodes there are avaricious men that needs more money and charitable men that feel the urge to give money. I think that you have no doubt which of them do what when one meet the other. The problem is that many times the transition occurs only if there is the possibility that the flow of money goes from who want to give them to who want them so you need a third agents to create the right environment and conditions for the transition. Furthermore the loss of money must be counterbalanced by something similar.
Then there are others type of men that simply don't want to give or take money we can call them noble men and then there are other men that can take money without actually feeling the need, they simply take them, I will call them hydron men.
Lemon batteries are in fact one of the more complex and debated battery you could examine. When you plug the zinc and the copper in your lemon we can say that we have three main active species: copper, zinc and acid (citric acid). Zinc is one of that metals we compared to a charitable man it tends to donate electrons: $$\ce{Zn <=> Zn^{2+} + 2e^{-} }$$ Copper however is a noble metal, physics explain this because it has the d-bands of the electronic structure filled, so it doesn't accept zinc's electrons. Instead hydrogen from acid, can accept the electrons in this way: $$\ce{2H+ + 2e^{-} -> H2 ^}$$ And this is in fact what actual experimental results show, see J. Chem. Educ. 2001, 78 (4), 516 for reference.
The short answer from the website Experiments in Electrochemistry (Carboni, 1998) is that
The Copper ($\ce{Cu}$) atoms attract electrons more than do the Zinc ($\ce{Zn}$) atoms.
According to the Naked Scientists website Lemon-Powered iPod - Fruit batteries, they state that
Zinc metal can dissolve to form zinc ions, releasing energy; it also loses electrons. If the zinc is connected to the copper by an electric circuit, these electrons can flow around the circuit and reduce copper ions in the lemon.
The reason why this occurs, is well described in the UC Davis Chemwiki Standard Reduction Potential: Activity Series, that as copper has a standard reduction potential of 0.34V, hence will be reduced (as described above), as zinc's standard reduction potential of -0.76V is considerably lower.
However, some copper ions dissolve into the lemon's citric acid (Naked Scientists).
The process is shown in the image below:
this continues until the electrons (according to the Carboni, 1998 article linked above):
concentrate on the copper, the electrons repel each other. When the force of repulsion between electrons and the force of attraction of electrons to the copper become equalized, the flow of electrons stops.
Some more explanation is available in the question and answer at Electronegativity of zinc vs copper in galvanic cell
