It'd help if you could clarify exactly what you're looking for - if you want specifics on the operational side of the war, I can elaborate, but for now I'll explain the circumstances of the war.
Ethiopia-Somalia relations were a problem from the end of WW2. Ethiopia's eastern regions consist of the sparsely populated Somali-majority Ogaden region and are a legacy of the first Italo-Ethiopian war. Although the Italians would restructure Ethiopia's borders during the occupation period, including Ogaden in Italian Somalia, the decision was made post-WW2 to return Ethiopia to its pre-war borders and award Italian Eritrea to Ethiopia with some special stipulations on governing it.
In the period leading up to Somali independence, the British hoped to forge a workable national identity among the population of the now merged Italian and British Somalilands by fostering Somali nationalism. This somewhat worked in the short term (although the same clan-based conflicts that plague Somalia to this day would wrack the country in its early years), but it created a problematic situation, as large Somali populations existed in neighboring Ogaden, Djibouti, and Kenya. When Siad Barre came to power in Somalia after a coup in 1969, he would leverage Somali nationalism and irredentist sentiment to secure his position. His nominally left-wing government turned to the Soviets for aid, and the Soviets were happy to oblige, providing a great amount of military aid in return for the creation of a naval base at the strategically important northern port of Berbera. Barre in turn used this military aid to further irredentist ends, sponsoring rebel groups in neighboring Kenya and Ethiopia.
This whole time, the regime of Haile Selaissie in Ethiopia had been slowly losing its grip due to Selaissie's advancing age and refusal to delegate powers to more competent subordinates. Despite the warning of a failed coup in 1961, Selaissie continued on with a variety of policies that would lead to unrest in his highly diverse domain, culminating in a successful coup in 1974. Like Somalia, this coup was also performed by left-leaning officers who established their own military government known as the Derg. The collapse of the imperial regime and subsequent power struggle among left-wing and ethnically-based groups in Ethiopia would lead to a slew of human rights abuses that rapidly alienated the US government (who up to this point had sponsored Ethiopia), while the Soviets jumped at the chance to gain another left-wing ally in the strategically important Horn of Africa. Leveraging Cuban support, the Soviets and Cubans worked to reconcile Ethiopia and Somalia to create a Soviet-aligned federation.
Unfortunately for the Soviets, Cubans, and most of all, Ethiopians, Siad Barre had no intention of following through with any reconciliation. Barre watched the rapidly deteriorating situation in Ethiopia and gambled that the time was ripe to seize Ogaden before Soviet support had really taken hold in Ethiopia. Unfortunately for Barre, the gamble didn't pay off. Castro was reportedly furious to hear about the invasion, as he had been working hard to convince Ethiopia and Somalia to form a federation, and the Soviets began to slow their aid to Somalia in hopes of convincing Barre to withdraw. The Soviet supplies slowed to a trickle and finally stopped just as Somali forces got bogged down on Ethiopian opposition short of their main strategic objectives, so Barre gambled once more.
In a desperate attempt to find a new source of supply for his war, Barre ejected the Soviets from the country. Around 2,000 Soviet naval personnel, their families and all Soviet citizens were given 48 hours to leave the country and their recently completed naval base at Berbera. The hope was that shutting down this controversial base would gain the support of the US and other anti-Soviet nations (Yugoslavia, Communist China, and some Arab states). Unfortunately, the US wasn't particularly keen on supporting a country waging an unprovoked war of aggression, and other countries with fewer qualms on the matter lacked the means to get the supplies to Somalia as fast as they needed it.
Unsurprisingly, the Soviets were less than enthused by this. They threw their full weight behind Ethiopia, launching their largest airlift and sealift effort since WW2. Planes were landing in Addis Ababa as fast as they could be pulled off the runways, weapons and aircraft were diverted from Yemen and other Eastern Bloc states to Ethiopia, and 50,000 Cubans were flown in to fight on the front.
Months of preparation followed, and the Ethiopian/Cuban force launched their counterattack - coincidentally on the same day the Somalis tried to renew their offensive. The now hopelessly outmatched Somalis were halted almost immediately and thrown back from their positions. The Ethiopians led the front, pushing the Somalis back to a rear base at Jijiga. From there, the Cubans airlifted men and armor around the rear of the Somali positions (marking the only airborne armored assault in history) and the Somalis were routed. From there, the remainder of the war was mopping-up actions to chase the remaining Somali forces out of their country.