What is a fan-in of a flip flop?
Fan-out is the number of logical inputs an output can drive. (Both Kaz's answer as well as the Wikipedia entry about it are wrong.)
Even for HCMOS fan-out is expressed as the number of LS-TTL inputs it can drive. The input current of an HCMOS gate is negligible.

Fan-in is a rarely used term simply meaning the number of inputs. 1 input has a fan-in of 1. Silly, but it fits with the fan-out definition.  

edit
ravi says this in his answer (from Wikipedia): 


  "Physical logic gates with a large fan-in tend to be slower than those with a small fan-in, because the complexity of the input circuitry increases the input capacitance of the device."


This isn't true. More inputs doesn't necessarily mean a more complex circuit. An 8-input NAND gate has the same input capacitance per pin as a 2-input one. Each input is connected to two MOSFETs, one P-MOSFET and one N-MOSFET.  



If an input has to drive a large number of FETs it will usually be buffered. In a microcontroller an I/O pin will probably only see two FET gates.

edit (upon request by clabacchio)
To clarify: more complex ICs may be slower, but it's not because of increased input capacitance as the Wikipedia article claims.
This is the logic diagram of a 74HC30:  

  

Notice that each input only sees one complementary pair. You'll see this in most logic ICs.