How do I change the password for my registered nick?

First you need to identify for your nick using your existing password. See the above question if you're unsure how this is done. Once identified, type:

/msg nickserv set password <new-password>

This will change your old password to whatever you stipulate for <new-password>. The change will take effect immediately.

Example:

/msg nickserv set password n3wsecret