The Constitution of the United States says that Congress is the only body in the United States that has the right and the power to declare war. Even so, the US has been involved in many conflicts without a formal declaration of war. In fact, Congress has only declared war five times (though there have been declarations of war against multiple countries in some of those instances) in the history of the United States.
The first of these was in 1812. This was when Congress declared war on the United Kingdom. We call this war the War of 1812.
The second time that Congress declared war was in 1846. In that year, Congress declared war on Mexico. This is typically called the Mexican-American War.
The next declaration of war came in 1898. In that year, Congress declared war on Spain. This war is referred to as the Spanish-American War.
Nineteen years later, Congress declared war twice. It declared war on Germany and later in 1917 on Austria-Hungary. Both of these declarations were part of World War I.
The final time that the Congress has declared war was in 1941 and 1942, when the country entered World War II (and when it declared war against additional countries). Congress declared war against Japan first, then against Germany and Italy a few days later, and finally against Bulgaria, Hungary, and Romania a few months later.
Since then, the US has participated in such wars as the Korean War, the Vietnam War, and the Persian Gulf War, all without formal declarations.