What is the standard format for a browser's User-Agent string?

The User-Agent header is part of the RFC7231, which is an improved version of the RFC1945, where it states:

The User-Agent request-header field contains information about the user agent originating the request. This is for statistical purposes, the tracing of protocol violations, and automated recognition of user agents for the sake of tailoring responses to avoid particular user agent limitations. User agents SHOULD include this field with requests. The field can contain multiple product tokens (section 3.8) and comments identifying the agent and any subproducts which form a significant part of the user agent. By convention, the product tokens are listed in order of their significance for identifying the application.

   User-Agent     = "User-Agent" ":" 1*( product | comment )

Where product is defined as:

   product         = token ["/" product-version]
   product-version = token
   token           = 1*<any CHAR except CTLs or separators>

And comment as:

   comment        = "(" *( ctext | quoted-pair | comment ) ")"
   ctext          = <any TEXT excluding "(" and ")">

This is specified in RFC 1945 in the section on Request Headers. It is not a very standardized format, though, and user agents tend to put whatever they want in there.

Yes, see: mozilla website, but as it was mentioned before. Basically you can put whatever you want there. For statistical/analytical purposes, the most important thing is, that every browser/os should have this standardized for itself.