Spontaneous emission (SE) rate and the fluorescence efficiency of a bare fluorescing nanoparticle and the nanoparticle with a silver nanoshell are analyzed rigorously by using a classical electromagnetic approach with the consideration of the nonlocal effect of the silver nano-shell. The dependences of the SE rate and the fluorescence efficiency on the core-shell structure are carefully studied and the physical interpretations of the results are addressed. The results show that the SE rate of a bare nanoparticle is much slower than that in the infinite medium by almost an order of magnitude and consequently the fluorescence efficiency is usually low. However, by encapsulating the nanoparticle with a silver shell, highly efficient fluorescence can be achieved as a result of a large Purcell enhancement and high out-coupling efficiency for a well-designed core-shell structure. We also show that a higher SE rate may not offer a larger fluorescence efficiency since the fluorescence efficiency not only depends on the internal quantum yield but also the out-coupling efficiency.