A website requires a user to create a password consisting of one lowercase letter from a 26-letter alphabet followed by two digits. If no digits may be repeated, then how many different passwords are possible?

This is an example of the fundamental counting principle. It says that if there are m choices for the first task and n choices for the second task, then there are m*n choices for both tasks. This principle can be extended beyond just two tasks. 
In this case, we have three spaces to fill in for the password. The first space must be a letter of the alphabet, so there are 26 choices for the first space. The second space can be any digit from 0-9. So for the second space there are 10 possible choices. 
Now for the third spot, we must also use a single digit. But we are not allowed to repeat the digit used in the second space. So we had 10 choices for the second space, but once a number is used there, it is no longer a viable choice for the third space. Thus our choices for the digit in the third space is reduced to 9.  
Now, as the counting principle states, we must multiply all our choices together. So the total number of possible passwords is:
26*10*9=2340