And I'd like to ask you about second names. Explain, what does Ann mean in her name? Her name's Alice Ann Bailey. What's that Ann for?
In here, second name means the father's name. For ex, Helena Petrovna Blavatsky. This Petrovna is Petr-ovna: Petr (Peter) was her father's name, and -ovna is a female suffix for belonging to a certain father. But in English countries I see: Alice Ann Bailey, John Richard Smith etc. People wear both male and female "father's" names!? I don't understand.
Thank you.
I won't answer the Theosophical questions as I don't know enough to comment intelligently. I can, however, respond about how English names are given and what they tend to mean.
Traditionally in English cultures (the UK, Canada, the US, and I assume also in Australia) children usually take their father's family name as their last or surname. In the example of Alice Ann Bailey, we can assume that her father's family name was Bailey. When a woman marries though she often drops her surname and takes the surname of her husband. So Bailey might not have been her father's family name but rather the family name of her husband.
With the growth of feminism in the 1960s some women chose to keep their original surnames even when they married, so a wife might end up with a different last name than her husband. Some opted to add their husband's last name to their own, with a hyphen (i.e. a woman who was originally a Smith who gets married to a man named Brown might change her name to Smith-Brown.) Sometimes the wife doesn't change her name at all when she gets married, but they choose to use hyphenated names for their children -- for example a woman named Smith married to a man named Brown might give their children the surname of Smith-Brown.
Names other than the family or surname are really just given according to the whims of the parents. Often children are given names in honour of favourite family members. In my own case, one of my names, Thomas, was given to me in honour of my father's dad, my grandfather. One of my brothers was given a name after my mom's dad. In all these cases though it was a first or middle name, not a surname, that was passed on. My brothers and I, like many kids in English countries, carry only our father's surname.
So in the case of Alice Ann Bailey, the names Alice and Ann were given to her by her parents either because the parents just liked the sound of the names or because they wanted to honour a family member or friend who had one or both of those names.
According to the Wikipedia entry for Alice Ann Bailey, she was given the name Alice LaTrobe Bateman by her parents. (I don't know if LaTrobe was her mother's family name or not.) So presumably she herself chose to add in the Ann, likely at the same time when she assumed the name Bailey. I don't know where she got these names as the only husband I am aware of was a man named Walter Evans, who she divorced after spending eight years with him.
Alice and Ann are usually used as first names rather than as family names in English. And to make things even more confusing, sometimes people give their children first names that are based on family names (either family names they are biologically associated with, or just names that the parents thought sounded good.) For example, the current President Bush had a grandfather named Prescott Bush. Prescott is usually a surname rather than a first name, but it was used as a first name in the Bush family.