These three words are all nouns, the most common of which are vale and veil. Vail is a very little-known word, except as the name of a town in Colorado, famous for its ski-resorts.
Vale is an old-fashioned word for valley, dale, or hollow—a small valley. It is used mainly in older or more poetic writing and place-names. However, it is well-known from the phrase “vale of tears” describing human life—with the idea that life is full of suffering:
I hope to spend many more years in this “vale of tears”!
The phrase “vale of tears” was originally associated with Christianity, but not anymore.
A veil is a very thin piece of cloth, often made of lace, usually to cover the body or face. The veils worn by women getting married are called bridal veils and the cloths covering the faces of Muslim women can be called veils (or burqa). Any thin cloth hiding something can also be called a veil.
Veil is also often used poetically, in clichés such as in the following examples:
The experiment was covered by a veil of secrecy.
The scientist tried to look behind the veil separating life and death.
They traveled under a veil of darkness.
And veil can be used as a verb, although this is less common:
The government veiled the project in secrecy.
As for the word vail, which is almost never used, it means “to repeatedly lower something as a sign of respect.”