What is the internal difference between rechargeable and normal batteries?

The critical difference is the chemical makeup of the paste electrolyte in the units by design. Some chemical reactions are reversible by adding current in reverse polarity (when charging), some are not.

The specifics of the compounds used in common batteries are covered in high school level chemistry books and the uncommon types are secret (proprietary) to the companies that manufacture them.

Many, many books have been written if you choose to search for them in the library or used book stores. Museums often have displays with historical types of cells or batteries.

Here is one useful book about lead acid batteries - the common ones found in automobiles.

http://www.lindsaybks.com/bks2/lead/index.html