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).