The HD44780 (and compatible) based alphanumeric displays are usually available in 16x2 (2 lines consisting of 16 characters) and 20x4 configurations, but can be found in many more forms. The most 'complicated' display would be a 40x4 display, this sort of display is special as it has 2 HD44780 controllers, one for the upper two row and one for the bottom two rows. Some graphic LCDs have two controllers as well.
The HD44780 LCDs are great, they are very cheap, readable and quite easy to work with. But they also have some drawbacks, these displays take up a lot of I/O pins when connected to the Arduino. In simple projects this is not a concern, but when projects get big, with a lot of IO, or where certain pins are needed for things such as an Analog read or PWM, the fact these LCDs require a minimum of 6 pins can become a problem.
But we can solve this problem in a cheap and interesting way.