anniversary
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anniversary
The day that commemorates an important personal event that recurs on that date, such as a wedding anniversary or the start of a relationship.
Morgen feiern wir unseren zehnten Jahrestag.
Tomorrow we are celebrating our tenth anniversary.
The word is clearly composed of 'Jahr' (year) + '-es-' (linking s) + 'Tag' (day). So it literally refers to the day on which an event occurs again for another year.
Der Jahrestag ihrer Hochzeit ist im Mai.
The anniversary of their wedding is in May.
An ordinal number is very often used to indicate which anniversary it is. For example: 'der erste Jahrestag' (the first anniversary), 'der zehnte Jahrestag' (the tenth anniversary).
Das Paar feiert heute seinen fünften Jahrestag.
The couple is celebrating their fifth anniversary today.
When it is about a personal event, 'Jahrestag' (anniversary) is often used with a possessive pronoun such as 'mein' (my), 'unser' (our), or 'ihr' (her/their). Typically, an anniversary is 'gefeiert' (celebrated).
Wir gehen schick essen, um unseren Jahrestag zu feiern.
We're going out for a nice dinner to celebrate our anniversary.
With historical events, the anniversary is often connected with a genitive construction: 'der Jahrestag des/der...' (the anniversary of...). Here the verb 'gedenken' (to commemorate) often fits better than 'feiern' (to celebrate).
Die Stadt gedenkt des Jahrestages ihrer Gründung.
The city commemorates the anniversary of its founding.
birthday
'Geburtstag' and 'Jahrestag' both refer to a day that reminds people every year of an earlier event. However, 'Geburtstag' is connected only with a person's birth. 'Jahrestag' is more general: it can commemorate a wedding, the start of a relationship, an accident, or a historical event. So a birthday can be a kind of anniversary, but it is not normally called that in German. For people and celebrations, you say 'Geburtstag'; for other important dates, you are more likely to say 'Jahrestag'.
Der Jahrestag ihrer Hochzeit ist für Eva und Paul sehr wichtig.
The anniversary of their wedding is very important to Eva and Paul.
Tara schreibt im Bus eine Nachricht, weil ihr Bruder heute Geburtstag hat.
Tara is writing a message on the bus because it is her brother's birthday today.