what is BES & BIS on the BB?

BES = BlackBerry Enterprise Server (i.e., Corporate BlackBerry server)
BIS = BlackBerry Internet Service (i.e., Personal BlackBerry service)

If you have a BES data plan, you can attach to a BES and also use BIS (if the BES policy allows it).

If you have a BIS data plan, you can only use a BB for personal use and cannot connect to a BES.

Do either of these save internet searches to the internal memory?

Both of them save your web history similar to computer browser programs.

No need to keep asking the same question: Internal memory on 8310 curve