So long as the broken off fan blade is completely removed and isn't rattling around between the fan and heatsink, you'll be fine for now. If not, then there is the possibility of it causing the fan to jam and therefore your CPU to get hotter due to the reduced efficiency of the heatsink. After-market HSF assemblies are available and are recommended for those who wish to do serious overclocking, although you don't have to overclock to use one. Simply ensure that the one you get is suitable for the mounting on the motherboard and that the fan has the same number of wires (3 or 4). Better still, remove the old HSF and take it with you so that you can get the correct type.
First of all, remove the old HSF.
When you have removed the old heatsink, make sure that you thoroughly clean off all residual TIM from the top of the processor. Remember that yours will be on the board, and not as shown here, but the principle is the same.
If your new HSF comes with pre-applied TIM, then skip this section. If not, you will need to apply this yourself. In this case, you apply it to the processor. This shows an Intel processor, but it's exactly the same method for an AMD one.
Now, fit the replacement HSF. If the HSF came with pre-applied TIM, remember to remove the protective covering before fitting.
And that's it. Don't forget to ensure that the fan is plugged into the motherboard.
HSF : HeatSink and Fan assembly.
TIM : Thermal Interface Material.