I posted this answer on the 4.75 digit topic4.75 digit topic, but I thought it would be useful here too.
Background
Digits is an old way of describing the resolution of a digital multimeter. The meaning of half-a-digit is pretty well established - it can only display the value "1". Problems started when manufacturers came out with things like 4.25 digits or 4.75 digits.
Since there is no standard as to what the 0.25 or 0.75 mean, manufacturers are free to come up with their own interpretation, which just causes confusion for everyone. For this reason, vendors started using counts.
A number of counts is the maximum value the multimeter can display, plus one. For example, a 4000 count meter can display values up to 3999.
The answer
Let's start with the most standard option of 0.5 count. A 4.5 digit meter would be capable of displaying at least 20,000 counts (max value of 19999). It means you have four full digits and one half digit, only capable of displaying 1.
The 4.75 digits is a more of a marketing term and could mean anything between 30,000 and 50,000 counts. Note that the Fluke 187, which is a 50,000 count meter is still described as 4.5 digit meter by Fluke themselves.
But wait! Then there's also 4.25 counts. Does your meter have 4.5 digits, but does not go all the way to 19999? No worries, just describe it as 4.25 digit multimeter! See this Aim-TTi 1705 spec sheet as an example. It is a 12,000 count meter, by the way.