Carolingian Empire is a historiographical term sometimes used to refer to the realm of the Franks under the dynasty of the Carolingians. This dynasty would later be seen as the founders of the Holy Roman Empire. As such this Empire can be seen as the later history of the Frankish Realm or the early history of the Holy Roman Empire depending on one's perspective.
When used, the term emphasizes on the coronation of Charlemagne as Emperor in 800 by Pope Leo III, although this did not actually constitute a new empire because both Charles as well as his ancestors had been rulers of the Frankish realm earlier. In fact, his grandfather Charles Martel had essentially founded the empire during his lifetime. Because of this, most historians prefer to use the term "Frankish Kingdoms" or "Frankish Realm" to refer to the area covering parts of today's Germany and France from the 5th to the 9th century.
Similarly the Empire did not truly "end" in a historical sense with the end of the dynasty since the Holy Roman Empire continued for several more centuries. Nevertheless, historiographically the "First Holy Roman Empire" can be said to have ended with the death of the Holy Roman Emperor Charles III the Fat in 888, although some Carolingians managed to gain the Imperial crown in later times.

Buildup and defense of the Frankish Realm
The Carolingian Empire at the death of Charlemagne covered most of Western Europe like the Roman Empire had. Unlike the Romans, who had rarely ventured beyond the Rhine after the disaster at the Battle of the Teutoburg Forest (9 AD), Charlemagne crushed all Germanic resistance and extended his realm completely to the Elbe, and influenced events almost to the Russian Steppes.
The Empire of the Carolingians had been divided among various members of the Carolingian dynasty. From the inception of the Empire, these included: King Charles receiving Neustria, King Louis the Pious receiving Aquitaine, and King Pepin receiving Italy. Pepin died with an illegitimate son Bernard in 810, and Charles died without heirs in 811. Although Bernard succeeded Pepin as King of Italy, Louis was made co-Emperor in 813 and the entire Empire passed to him with Charlemagne's death in 814.

The Empire after the Treaty of Verdun (843 - 877)
Charles the Bald died in 877 crossing the Pass of Mont Cenis, and was succeeded by his son, Louis the Stammerer as King of the Western Franks, but the title of Holy Roman Emperor lapsed. Louis the Stammerer was physically weak and died two years later, his realm being divided between his eldest two sons: Louis III gaining Neustria and Francia, and Carloman gaining Aquitaine and Burgundy. The Kingdom of Italy was finally granted to King Carloman of Bavaria, but a stroke forced him to abdicate Italy to his brother Charles the Fat and Bavaria to Louis of Saxony. Also in 879, Boso, Count of Arles founded the Kingdom of Lower Burgundy in Provence.
In 881, Charles the Fat was crowned the Holy Roman Emperor while Louis II of Saxony and Louis III of Francia died the following year. Saxony and Bavaria were united with Charles the Fat's Kingdom, and Francia and Neustria were granted to Carloman of Aquitaine who also conquered Lower Burgundy. Carloman died in a hunting accident in 884 after a tumultuous and ineffective reign, and his lands were inherited by Charles the Fat, effectively recreating the Empire of Charlemagne.
Charles, suffering what is believed to be epilepsy, could not secure the kingdom against Viking raiders, and after buying their withdrawal from Paris in 886 was perceived by the court as being cowardly and incompetent. The following year his nephew Arnulf of Carinthia, the illegitimate son of King Carloman of Bavaria, raised the standard of rebellion. Instead of fighting the insurrection, Charles fled to Neidingen and died the following year. The Empire of the Carolingians was divided: Arnulf maintained Carinthia, Bavaria, Lorraine and modern Germany; Count Odo of Paris was elected King of Western Francia (France), Ranulf II became King of Aquitaine, Italy went to Count Berengar of Friuli, Upper Burgundy to Rudolph I, and Lower Burgundy to Louis the Blind, the son of Boso of Arles, King of Lower Burgundy.