All batteries use three ingredients: two electrodes (an ANODE and a CATHODE) and an ELECTROLYTE in between.
At the ANODE, the material loses electrons, generating positively charged ions.
Losing electrons? That's OXIDATION.
At the CATHODE, the material collects electrons, generating negatively charged ions.
Gaining electrons? That's REDUCTION.
The negative ions at the Cathode move through the ELECTROLYTE to the Anode
... and through an external wire from Anode to Cathode (thereby lighting a lamp, for example).

This so-called REDOX reaction occurs between many types of materials used for the Anode, Cathode and Electolyte.
The voltage produced depends upon what material are used.

Example: Zinc & Copper & Sulphuric Acid

Stick a zinc rod into a jar of sulphuric acid.
The sulphuric acid molecules H₂SO₄ breaks up into charged ions: two Hydrogen H⁺ ions and a Sulphate ion SO₄^-
The Zinc (anode) breaks up into zinc ions Zn⁺⁺, releasing two electrons e^-. This release of electrons is OXIDATION.
Zinc ions Zn⁺⁺ combine with a Sulphate ions SO₄^- to create Zinc Sulphate: ZnSO₄
The electrons released at the Zinc anode desperately want to combine with some positively charged ions.
The electrons released at the Zinc anode then combine with the two Hydrogen H⁺ to produce hydrogen gas. This gain of electrons is REDUCTION.
Unfortunately, all this Oxidation-Reduction all takes place at the zinc anode.
If we could have the electrons travel through a wire in order to achieve Reduction, the current could do some work (like light a lamp).
To achieve this, we put our Zinc rod and sulphuric acid into a Copper jar.
Copper is not oxidized by the acid so the the H⁺ ions in contact with the copper cannot receive electrons from the copper. Neither can they receive electrons from the Zinc, because they are not in contact with the Zinc.
However, connecting the Zinc anode to the copper jar with a wire allows the electrons produced at the Zinc to travel throught the wire to the Copper.
The H⁺ ions at the Copper can then collect their electrons. They're happy. The negatively charged SO₄^- ions in the acid, at the Copper surface, are left without their associated hydrogen ions (which have been reduced by collecting electrons at the Copper), so they migrate through the acid to the positively charged Zinc ions Zn⁺⁺ at the anode.

Example: in general

The Anode gives up electrons, leaving positively charged ions: Oxidation.
The electrons travel through the wire (lighting a lamp, for example), then renter the battery at the Cathode.
At the Cathode the electrons (arriving through the wire) combine with the cathode material to produce negatively charged ions: Reduction.
The negatively charged ions at the cathode flow through the electrolyte to the anode to combine with the postively charged ions there.

Other materials:

LEAD & LEAD OXIDE & sulphuric acid in between (the car battery)
ZINC & CARBON & electrolitic paste in between (ordinary flashlight batteries)
NICKEL HYDROXIDE & CADMIUM & potassium hydroxide in between (the rechargeable NiCad batteries)
NICKEL & METAL HYDRIDE & ?? in between (rapidly replacing NiCads)
LITHIUM compound &