What is the keychain? And why do I get these nagging keychain pop-ups?

Keychain Access icon  
The keychain is probably one of the least understood features on the Mac, yet it is probably one of the most useful.  The keychain is analogous to an actual keychain.  A keychain is used to hold your keys.  On the Mac the keychain stores your many passwords.
 
To look at what it's saved for passwords, you need to go to the Keychain Access application in your Mac's Utilities folder.  The easiest way to open Keychain Access is to use Spotlight and search for it...
 
Searching for Keychain Access using Spotlight  
 
Press the Return key to select the Top Hit. This will open the Keychain Access program.
 
In Keychain Access, a window lists all of the keychains your Mac has saved.
Keychain Access window  
 
You can double click on any item in the list and click the check box to show the password that the keychain remembers.  You will be asked for your computer password.  You must enter it correctly for the password to be revealed.  It's asking for a password to see a password — based merely on security in the event that someone is using your computer without your permission.
Specific keychain item  
 
I'm getting nagged for a keychain password and it won't go away.  Make it stop! 
In the event that your getting bombarded for a keychain password and no password you try works, it's probably best to delete your Login keychain from Keychain Access and logout of the account.
 
With Login selected from the Keychain Access side bar, go to File > Delete Keychain Login.
Deleting Keychain "login"  
 
When you log back into the computer a new login keychain will be created.  If you've changed your computer password and are prompted a login about keychain issues, it's probably best to choose "Create New Keychain."  This avoids any and all keychain pop-up issues that may occur.