I know that 'V' in the German alphabet is for "Vau" and 'F' is "Eff".
Then why are we using 'V' instead of 'F' in German for "Vater"? Similar to the English "father" I would have expected it to be spelt "Fater" rather than "Vater".
In all languages including German spelling evolved over time with no fixed rules on ortography or spelling. Nevertheless people tried to find letters for phonetically similar sounds. In the family of phonetically related letters for the modern 'F' we can find the following, also relevant for "father", and "Vater" which have a common Indo-European root with Latin "PATER":
These letter were at times used interchangeably, probably also depending on regional dialects. In Old High German the letter 'V' was not yet used for the phoneme 'F', thus father indeed was spelt "fater". The letters 'U and 'V' were initially identical, and the letter "W" was built from doubling "UU".
In Middle High German the letter 'F' was continuously being replaced by 'V', making this letter far more frequent than today. Many words which today are spelt with 'F' were written with a 'V' then (e.g.: Vrouwe - Frau, Vackel - Fackel, Valke - Falke, vlicken - flicken and many many more).
For reasons read: Warum wurde im Mittelhochdeutsch 'F' durch 'V' ersetzt? (German)
From the 14. Century however writers in the chancellery of the Luxembourgian-Austrian monarchy, later also in the archives of Frederick III, Holy Roman Emperor, and in the literature the letter 'V' was increasingly replaced again by the letter 'F'. This was not done entirely. Some exceptions were still spelt with 'V'. A major influence on these spellings also came from Martin Luther's translation of the Holy Bible, where e.g. father is uniformly spelt "Vater".
Today, only few remnants of the Middle High German spellings are left:
From this rather arbitrary approach how spelling has developed there is no fixed rule to aid us here. We all have to and had to learn them by heart. Many German pupils will face problems with the "Vogel Vau" when learning to write.
For more details on the letter 'V' see: Grimms Wörterbuch
I haven't a good explanation for this, especially as it looks like the word was actually spelled with "F", e.g. in the Hildebrandslied, one of the oldest German documents:
As far as my source knows, it comes form the medieval.
All these letters developed from the semitic alphabet "waw", which was used for /w/ and /u/, via Greek and Latin. In Latin, the letter v still had these two ways of pronounciation.
Then happened language development in the European languages (see Takkats Answer for more information). Sounds shifted, scholars influenced the writing by different intentions, languages diversified and intermixed, …
The result in German was: