Crew Resource Management started out in the 70s as an attempt to eliminate human error but over time it's undergone a number of iterations and today the concepts have accepted reality and assume errors can't be eliminated, only mitigated.
Modern CRM tries to mitigate errors by attempting to require multiple errors to stack up in an improbable way before things become dangerous (called the "swiss cheese" approach, as in mutiple layers of swiss cheese where a single hole is unlikely to line up). It also has added personality factors, human interaction factors, and leadership factors to the mix.
In practical terms, this means that, for example, captains are expected to act as leaders and not bosses. The FO is not just a gear puller and is encouraged to participate in assessment, planning and decision making. Copilots are expected to speak up when in doubt about something. Both crew members are expected to make sure that each is aware of the other's actions, and planned future actions, at all times. Crew members cross check each other's actions continuously. Crew members learn standard phraseology for key actions and are expected to stick to it to reduce confusion. Crew members are expected to follow standard operating procedures and not invent their own ways of doing things.
ATC gives an altitude clearance change? Pilot monitoring and talking to ATC dials the autopilot to the new altitude, and calls it out something like "Flight Level 350 set". Pilot flying confirms the correct altitude is set and calls out "350 checks". You generally announce everything you do as you do it and make sure your partner is aware (usually by responding "check"). It should just flow. One major bonus is the standardization makes it possible for crews to smoothly interact without flying together for a period of time first.
A cockpit with good CRM will have a well-oiled machine aspect to it with the pilot and copilot well integrated mentally. The capt and FO cooperate on planning and decision making, notwithstanding the fact that it's the captain's final call. They will look like they've been flying together for weeks when they've actually just met that morning.
I would say that the two biggest factors in the amazing safety record of airline transport in the last 30 years, from a crew performance standpoint, is the success of CRM, and the advent of the 2D moving map presentation when the glass cockpit came along, which massively improved situational awareness by eliminating the need to generate a mental picture of a navigation situation like in the steam gauge era.