Learn Russian: Я вижу экран, а у меня нет пароля. - I see the screen, but I have no password.

This is the core of this lesson! When you use нет to say you don't have something, the thing you don't have must be in the Genitive case. The dictionary form is пароль (a masculine noun ending in a soft sign). In the Genitive case, that soft sign changes to -я, giving us пароля.

How do we know to use а for 'but' here instead of но?

The word а acts like a 'soft but' or 'whereas'. It is used here to contrast two related facts: on one hand, you see the screen, but on the other hand, you don't have the password. Но is stronger and usually implies an unexpected obstacle or direct contradiction.

Why doesn't экран change its ending? Doesn't the verb вижу (I see) force the word into a new case?

It does! The verb видеть (to see) requires the Accusative case for the object you are looking at. However, экран (screen) is a masculine inanimate noun (meaning it is not a living being). For masculine inanimate nouns, the Accusative case looks exactly the same as the basic dictionary form.

I learned that у меня есть means 'I have'. Why is the word есть missing here?