It depends on your meaning of the term concentration camp.

Both the British and Nazi's have used the term in the past, and the Nazi term is actually taken from the British.

The British concentration camps were used in the Second Boer War. Initially designed as refugee camps, eventually the idea progressed into the concentration camp.

Nazi concentration camps are similar to their British counterparts, but do not think concentation camp and extermination camp are the same thing. A extermination camp is a type of concentration camp focused solely on extermination of a population.

POW camps and labour camps are also both examples of concentration camps.

tl;dr - A straight answer to your question is The Second Boer War.

EDIT: Upon further reading the design of a concentration camp may have been used earlier, but the term was popularised by the British. Earliest mention of the actual term is by the Spanish in the Ten Years War as reconcentrados (reconcentration camps).

US Civil War - first concentration camp (by function, not by name) was Andersonville - it was called both Andersonville prison, and a Camp Sumter, but it acted effectively as a POW concentration camp, with similar organization.

Here's Andersonville survivor and Andersonville plan

Of the nearly 45,000 prisoners that came to Andersonville, more than 13,000 died - see 4.

Sources

1. James K. Polk - Horrors of Andersonvile

2. Ransom Chadwick - An Inventory of His Andersonville Prison Diary at the Minnesota Historical Society

3. Kellogg, Robert H. Life and Death in Rebel Prisons. Hartford, CT: L. Stebbins, 1865

4. The Horrors of Andersonville: Life and Death inside a Civil War Prison Camp by Catherine Gourley - Gouley calls it a Prison Camp.