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I mean to ask what do ions actually do while conducting electricity?
Do they simply take up the electrons and transport it? If it is so, I do not understand how anions can take up more electrons if they are already negatively charged? And if cations take up the electrons, won't they be converted to their elemental forms? Then they no longer remain an ion, so how will they keep conducting? Or is it a completely different mechanism?
There are never neutral atoms in pure $\ce{NaCl}$ or in its solutions. The pure $\ce{NaCl}$ crystal is made of piles of ions $\ce{Na^+}$ and $\ce{Cl-}$. When $\ce{NaCl}$ is dissolved into water, the ions $\ce{Na^+}$ and $\ce{Cl-}$ are separated. They become independent and can move in water. If you dip in this solution two conducting pieces bound to the poles of a battery, a positive pole (anode) and a negative pole (cathode) are created in the solution. The positive pole (anode) attracts the negative ions (anions) like $\ce{Cl^-}$ : they slowly migrate in the solution up to the anode, and when they touch the anode, they lose their charge and create a free electron which enters the anode and the external circuit. The anions become neutral atoms $\ce{Cl}$. So the anions do not take up electrons as you say. They already have electrons in excess, before the beginning of the electrolysis.
In the circuit "Negative pole of the battery - Electrode - Solution - Electrode - positive pole of the battery", the electric current is made of charge carriers which are electrons between battery and electrodes, and ions in the solution between the electrodes. The nature of the electric currant changes along the circuit.
To be correct I should also mention that positive ions like $\ce{Na^+}$ are also migrating in the solution, as they are attracted by the negative pole (cathode). So in the solution, there is one currant of positive ions moving to the negative electrode. And there is also a currant of negative ions moving to the positive electrode. As a result the solution remains neutral, without any positive or negative electric charges in excess.
In ionic compounds electron is transferred from one atom to another completely to form cation and anion, and resultantly, an ionic bond. To bring physics in this question, you have to understand that current is the flow of charge. All ions are charged. So positively charged ions move towards the negative electrode and anions move towards the positively charged electrode. As ions are moving, the charge is moving. This movement of charge produces current. However, Maurice has clearly answered your question and I would have upvoted, but my reputation doesn't permit that. My aim is to provide supplementary information regarding what ions do to conduct electricity as was your question.
