The reconstructed Proto-Indo-European form listed is *h₁wegʷʰ- "to promise, vow, praise".
- h₁ is a "laryngeal consonant", so-called because we don't know what else to call it. The most common theory is that there were three consonants that acted similarly, and since nobody knew what they were, they were written as h with a subscript. h₁ was most likely either like Latin h, or a glottal stop.
- w is a "consonantal u", like Latin v.
- e is a short vowel probably like Latin e.
- gʷʰ is the weirdest one. The standard reconstruction of PIE involves voiced aspirated consonants, but these are not particularly common, and this analysis leads to several other very strange features (such as the lack of a b sound). Other theories have been put forth but they have their own flaws. So in the standard theory, gʷʰ is like Latin qu, but voiced and aspirated.
- And the hyphen at the end means this is just a stem.
One particular form of this verb was the causative, *h₁wogʷʰ-éye-.
- The e became an o, which is a common though not entirely understood occurrence in PIE: consider Greek leg-ō and log-os. It's generally called Ablaut because it was first noted by German linguists.
This led to a Proto-Italic form *wogʷ-e-ō.
- The laryngeal disappeared entirely, which they tend to do.
- The gʷʰ lost its aspiration, which regularly happened in Italic.
- -ō was a standard verb ending, as seen in Greek and Latin.
And this became Latin vov-eō.
- gʷ became plain w, written as v in Latin. This was again a regular process.
Another form of this verb was *h₁ewgʰ-.
- The e and w switched places, a not-unheard-of process called metathesis.
- Then the gʷʰ lost its ʷ when right next to the w.
- This is called the Boukolos Rule, because the first example discovered was Greek βουκόλος = boukólos < *gʷou-kolos < *gʷou-kʷolos.
This became eukʰ- in Hellenic.
- The laryngeal vanished as usual.
- Greek kept the aspiration distinctions that Latin lost, but removed voicing from aspirated consonants.
- The consonant w became the vowel u.
In Greek, this was written eúkhomai (εὔχομαι).
- -o-mai is simply a thematic vowel + personal ending.