Lemon does not directly produce electricity. However, you can produce electricity using lemon and the right device. The idea behind it is simply the energy conversion: from the chemical energy to electric energy. This is actually the same principle for battery in our daily life.

Ok, back to the question, to produce the electricity, you need two different pieces of metal such as Zn (Zinc) and Cu (Copper) and a conducting wire like Cu as well. The acid inside the lemon are chemicals and has chemical energy and it can serve as the electrolyte. When the two pieces of metal are in contact with the acid, they are served as the electrodes. One of the electrodes will gain electrons and the other will lose electrons. When you connect the two electrodes, the electrons will flow along the conducting wire, that is the electricity.

There is a good website and a really good cartoon explaining the details:
lemon and electricity

Lemon juice contains citric acid, and so contains hydrogen ions. Electrons flow from the metal that has a higher electropositive potential (e.g. copper) to the metal with a lower electropositive potential (e.g. zinc). Meanwhile, to balance the charges, the positively-charged hydrogen ions in the citric acid, oxidizing the copper and reducing the zinc.