The default syntax for searching Web of Science is described on the "Search Rules" section of the Web of Science help page.
A more complete description of specific options used for BibDesk is given on the Advanced Search page and the Advanced Search Examples page.
If you just type a search term or group of terms in the search field, BibDesk will automatically convert it to a "Topic Search" (TS=), unless you specify the field tag manually. This is an easy way of searching Title, Abstract, and Keywords, but won't work for authors.
Apart from these searches, BibDesk supports a number of additional search operators to fetch associated items. These are `citedby:' for cited references, `citing:' for citing articles, `related:' for related items, and `uid:' to fetch one or more items by ID. These operators should be followed by a WOS UID, which is usually returned in the Wok-Uid field. The `uid:' operator accepts multiple UIDs. These operators cannot be combined with each other and with other search terms.
In addition, for all search types, you can add either or both of the search operators `begin:' and `end:' to restrict the search to a specific time span. They are followed by a date in the format Y-M-D, Y-M, or Y. Components can be omitted, in which case they are implied by the current date; e.g. `-3' indicates March of the current year.