Well, you just listed two of the four commonly recognized observations that a chemical reaction is taking place. A chemical reaction is when two or more substances react with each other, break electron bonds, establish new ones, and form new substances with different physical and chemical properties. When you combine the baking soda and the vinegar, the solution produced between the two bubbles and fizzes, indicating a gas is being produced. That gas is carbon dioxide. There is also a small amount of heat generated, indicating it is an exothermic reaction, which gives off energy in the form of heat.
Chemical reactions happen because of the elements within the compounds being involved willingness to break old bonds and establish new ones. This process produces both positively and negatively charged ions. When the ions are produced, you get a force of attraction between the two ions that goes together "just like peas and carrots", according to movie character Forrest Gump!
What actually happens is that the acetic acid (Vinegar) reacts with the sodium bicarbonate (Baking Soda) to form carbonic acid.
It is actually two reactions at once. Carbonic acid quickly breaks down into carbon dioxide and water. The bubbles are the carbon dioxide escaping from the water.
This combination can have many uses, From using the solution for a Science project to using it as a cleaning agent.