Aniversario vs. cumpleaños
The Spanish words 'aniversario' and 'cumpleaños' both denote some type of celebration revolving around a specific date, however, they are used in different contexts. The difference between the two lies primarily in the type of events they refer to.
Aniversario
'Aniversario' refers to the anniversary of a particular event. While this can include birthdays, 'aniversario' is more commonly used to commemorate other types of occasions such as wedding anniversaries or anniversaries of significant non-personal events.
Hoy es nuestro aniversario de bodas.
(Today is our wedding anniversary.)
Este año se celebra el 50º aniversario de la llegada del hombre a la luna.
(This year marks the 50th anniversary of mankind's arrival on the moon.)
Cumpleaños
'Cumpleaños', on the other hand, is specifically used to denote birthdays. It refers to the occasion of celebrating another year since someone's birth.
Hoy es mi cumpleaños.
(Today is my birthday.)
Vamos a celebrar el cumpleaños de Juan este fin de semana.
(We are going to celebrate Juan's birthday this weekend.)
Summary
In summary, while both 'aniversario' and 'cumpleaños' talk about commemorating important dates, 'aniversario' is generally used to denote anniversaries of all kinds whereas 'cumpleaños' specifically refers to birthdays.