In multiplier design an (n,k) counter takes an n bit input and produces a k bit output which is the binary representation of the number of input bits that are 1s. That is: it counts the number of input bits set to 1.
A (3,2) counter is just a full adder.
A (7,3) counter is a circuit with 7 input bits and 3 output bits. The 3 output bits tell you (in binary) how many of the input bits are set to 1.
In an (n, k) counter it must be the case that $k = \lceil \log_2 (n+1) \rceil$.
A (n:k) compressor is any circuit that takes in n equal weight input bits (plus, possibly, some carry in bits) and outputs a k bit count, plus some additional carry-out bits. The number of carry-in bits usually equals the number of carry-out bits, but you have to figure out the number by looking at the context. (Hopefully there's a picture or a detailed description in the paper or book you are reading.)
So a (7,3) counter is also a (7:3) compressor. But a (7:3) compressor could also mean something else, like something that takes in 7 input bits, plus a carry-in and outputs 3 output bits, plus a carry-out.
The most commonly used compressor is the (4:2) compressor. This takes in 5 inputs, and output 3 outputs. The reason it is not just a (5,3) counter is because the output is not a binary number in the range $[0, 5]$. Rather the output is a 2-bit value in the range $[0,3]$ plus an additional carry-out. So there are several ways of representing some of the outputs. For example "2" can be represented as "0+carry-out" or as "2+no-carry-out".