Learn Spanish (Spain): No recuerdo el nombre.

Recuerdo comes from the verb recordar (to remember). It is a stem-changing verb, meaning the o in the stem changes to ue when we conjugate it in the present tense for most subjects. Because the subject is "I", recordar becomes recuerdo.

Why doesn't the sentence say yo no recuerdo?

In Spanish, you do not usually need to include subject pronouns like yo (I) because the verb ending already tells us who is doing the action. The -o at the end of recuerdo makes it clear that "I" am the one who does not remember.

Does nombre mean "number"?

No, nombre is a false friend—a word that looks like an English word but means something completely different. Nombre actually means "name". If you want to say "number", the Spanish word is número.

Is it always just no before the verb to make a sentence negative?

Yes! To make a sentence negative in Spanish, you simply put no directly before the verb. So "I remember" is recuerdo, and "I do not remember" is no recuerdo. You never need extra helping verbs like "do" or "does" from English.