Instruments that have display counts specified are usually multimeters and component testers.  In both instrument types, there are different ranges that are used to obtain different values in measured readings.  Counts tell you what the instrument can display before it changes to the next range.

For example, suppose a multimeter has 50000 counts.  This means, the range changes when it hits 50000 on display.  To illustrate, a 50000 count meter can read 49.999 V on the display, but when it tries to display 50 V, it will only read  050.00 V, or simply 50.00 V instead.  The multimeter moves to the next range, which generally also means one digit of resolution will be lost.

Another example, suppose your LCR meter has 2000 counts.  0 to 19.99 can be read on display with two digits of resolution if you are trying to measure a component with 19.99 units.  But if the reading is to be 20, then the display will show 20.0 instead, losing one digit of resolution as it changes its range.  This is because the count is 2000, meaning when display hits 2000, it changes to the next range.