If you don't know what that is, Valve has some good documentation on it (relating to Source engine, but it's the same concept).[1]
Latency was still a problem even with CSP because we didn't yet have lag compensation on the server. So although you got an authoritative server (no cheating) and instant inputs (no round-trip wait), you could still shoot someone on the client and miss them on the server. You had to shoot ahead of them, further if your latency was worse. As far as I know, Source engine was the first to add lag compensation, which has the server rewind the other players to be where you would have seen them on the client.
Since client-side prediction can be performance intensive, and lag compensation can be difficult for anything beyond simple raycast checks, many games still don't do both. Team Fortress 2 for instance does lag compensation on basic guns but not on things like the soldier's rocket or the pyro's flamethrower. Some games simply allow the client to decide who they hit, which removes the need for either solution, but totally opens things up to cheaters without some very careful server-side validation.
[1] https://developer.valvesoftware.com/wiki/Latency_Compensatin...