Satellites are normally referred to by their NORAD ID numbers, which are assigned by the US military. There are various websites which will list these. Here’s a table from my own website…
The UN have their own separate satellite numbering system, which is the COSPAR ID you see in the second column. Not many people seem to use it, though.
I think when somebody refers to “element set satellite ID”, they’re refering to the “two-line elements” format used by websites such as Celestrak when publishing the current orbits of spacecraft. For example, their current data for the ISS is as follows…
1 25544U 98067A 20208.81898250 -.00001223 00000-0 -13780-4 0 9996
2 25544 51.6445 152.3082 0000916 163.1489 218.7089 15.49504233238167<
There’s full documentation on the Celestrak website about what all the numbers mean, but the second number on each line is the NORAD ID number of the spacecraft in question. The ISS has a NORAD ID of 25544.
Hope that helps,