Electric Battery

Some say that the battery supplies charge, some say that it only maintains potential difference, I've also heard that it maintains an electric field in the wire. I'm confused. What does battery actually do? Please tell me the reality.

Batteries do not "supply" charge to a circuit. They supply electrical potential energy to mobile positive or negative charge carriers already present in the circuit, by means of maintaining a potential difference across the terminals.

In the case of mobile positive charges (e.g., positive ions) they internally supply electrical potential energy to move the positive charge from a low potential location (the negative terminal of the battery) to a high potential location (the positive terminal of the battery).

In the case of a mobile negative charges (electrons) they internally supply electrical potential energy to move the negative charge form from a high potential location (the positive terminal of the battery), to a low potential location (the negative terminal of the battery).

In each case they do this by using chemical potential energy within the cell to perform the necessary work on the charge to give it electrical potential energy.

What happens when we connect anything with battery through the wire? Do the electrons are supplied by battery or electrons in the copper wire move? Do electrons flow through the battery? Please clarify. I need help!!

The electrons are supplied by the copper wire, not the battery. Metals like copper have highly mobile charge carriers, i.e., many free electrons. In the absence of an electric field (no battery connected across the wire) the free electrons move randomly because of thermal motion. When the wire is placed across the battery, the force of the electric field produced by the battery causes the electrons to collectively "drift" in a direction opposite the electric field, resulting in current (a.k.a. drift current). During their travel they lose electrical potential energy in the form of heat because they continually collide with the molecules of the wire along the way. When the electrons flow through the battery they regain their electrical potential energy from the battery to maintain the current.

Hope this helps.