Champagne is a sparkling wine, but not all sparkling wine is Champagne.
Champagne refers to a wine made using a specific process from a specific set of grapes grown in a specific region in France. The Champagne region is east of Paris, in northern France. Champagne is made from Chardonnay, Pinot Noir and/or Pinot Meunier.
The process to make Champagne is long and complicated. First, a standard base wine is made and bottled. Into each bottle, a combination of sugar and yeast is added and the bottle is sealed. The yeast eats the sugar, producing carbon dioxide and some additional alcohol. When the yeast has finished eating all of the sugar, it dies and begins to break down. These broken down yeast cells are called “lees." The lees are left in the wine for months or even years and add bread-y, toast-y notes to the wine.
During this lees' aging process, the bottles are slowly rotated and angled so they are neck-down. This process forces the lees to ultimately collect at the opening of the bottle. When the lees' aging is complete, the bottles are quickly opened to remove the collected lees and a small amount of wine is added to the bottle to fill the space. This added wine may contain some sugar to add sweetness. At this point the bottle receives its final cork and seal. The wine will then undergo some amount of bottle aging before being released for sale.
This elaborate process, known as the traditional method or methode champenoise, is used in other parts of the world, but the wine is not called Champagne. Cava from Spain, and some sparkling wines from other parts of France, the US, and even England, are made using this process. This method of making sparkling wine was actually developed in Limoux, in southern France, when the monks of Saint-Hilaire noticed that their wines would develop bubbles after being bottled.
Not all sparkling wines go through such an elaborate process. Prosecco, from Italy, starts similarly, but the secondary fermentation to create the carbon dioxide happens inside a large pressurized tank, instead of in the individual bottles. The large tanks do not provide as much lees contact, so the resulting wine tastes much fruiter than champagne and does not have as much of the bread-y, toast-y notes.
Some sparkling wines are made by simply injecting carbon dioxide, as is done with soda. This provides a much less expensive alternative to other methods, but the bubbles fade quickly leaving the wine flat.