This is traders’ jargon for the whole-dollar amount of a security quote. For example, if a security is quoted as $78.25 bid and $78.75 offered, the handle is 78.
In currency markets, the handle is the bit that is common to both the bid and offer. For example, if euro versus sterling (EUR/GBP) is quoted at 0.80771 bid and 0.80781 offered, the handle is 807.
Usually traders would not mention the handles when quoting prices, but just give the bit after the decimal point or the last two digits of a currency quote. That’s because it’s assumed that other traders know the handle.