The structure of a shift register is very similar to that of a counter. But while the counter generates new data at each clock, the purpose of a shift register is to store data. The circuit diagram of a shift register with D flip-flop is shown below.
With each clock signal Clk, the information Data is stored in Q3. Simultaneously, the information that was stored at that time in Q3, is now stored in Q2, the information from Q2 to Q1.... and so on. All information is shifted by one place at each clock signal.... hence the name shift register.
To properly understand how a shift register works, build it using the D flip-flops...