How can I replace a ceiling fan light pull chain?
Since you didn't provide a picture, or a very helpful description of what you're looking at. I'll try answering your question by explaining how the switch itself works, which will hopefully help you understand the problem better.

Single Pole Single Throw (SPST) Pull Chain Switch

The pull chain switch that controls the light(s), is a single pole single throw (SPST) switch. It has two positions ON (Closed), and OFF (Open). Drawn simply, it would look something like this.


Switch shown in ON (Closed) position.

When the switch is in the ON (Closed) position, current is allowed to flow through the switch, through the light(s), and back to the the source (via neutral).



When the chain is pulled and released, the internal contact rotates 90° (1/4 turn) into the OFF (Open) position.



When the switch is in this position, current is not allowed to flow through the switch, and the light is not lit.



This is why the pull chain switch that controls the light(s) only has two leads.

Single Pole Multiple Throw (SPnT) Pull Chain Switch

The pull chain switch that controls the fan, is a single pole multiple throw switch. It has multiple positions, which allows it to control the speed of the fan. Draw simply, it would look something like this.


Switch shown in OFF (Open) position.

When the chain is pulled and released on this switch, the internal contact rotates 45° (1/8 turn) to the next position.



Another pull, another turn.



Pull again, turn some more.



One final pull brings the switch 180° around, and again to the OFF (Open) position. 

By manipulating the output of this switch, the fan is able to whirl around at various speeds depending on the switches position. The number of output leads, will depend on the switch. How those leads are connected to the fan motor, will depend on the fan manufacturer.  This simply illustrates the basic principle of how the switch works.

As always electrical work can be dangerous, never be afraid to contact a qualified Electrician