You would rarely use one in real life. I believe the purpose of the Fibonacci heap was to improve the asymptotic running time of Dijkstra’s algorithm. It might give you an improvement for very, very large inputs, but most of the time, a simple binary heap is all you need.

From Wiki:

Although the total running time of a

sequence of operations starting with

an empty structure is bounded by the

bounds given above, some (very few)

operations in the sequence can take

very long to complete (in particular

delete and delete minimum have linear

running time in the worst case). For

this reason Fibonacci heaps and other

amortized data structures may not be

appropriate for real-time systems.

The binary heap is a data structure that can be used to quickly find the maximum (or minimum) value in a set of values. It’s used in Dijkstra’s algorithm (shortest path), Prim’s algorithm (minimum spanning tree) and Huffman encoding (data compression).