I like to think of our core as our torso from our neck down to our pelvis. Many people think of this as our abdominals, but there are many other muscles that comprise the core. The core is made up of both the inner core and the outter core. In order to create the greatest amount of stability while we are executing an exercise, we must engage our inner core and extend it out to our outter core, in the act called bracing. The main function of the core is to take on predictable as well as unpredictable stimulus in order to ensure the greatest amount of stability is created. The stability allows our posture to remain fixxed and allows our synergy within our musculoskeltal system to work as one.
If your core is not strong enough to execute a certain lift, your posture and form will be comprimised. When our form becomes compromised, we open ourselves up to become injured. This is what I mean by, "You are only as strong as your core allows you to be." If your core is not strong enough, your control of the exercise throughout the range of motion will put demands on muscle groups that should not be firing. A good example of this is in the standing tricep press down or extension. Many people will hunch over and round their upper back, trying to push down more weight than their core can support. I guarantee you, you will get a better benefit from the exercise by standing up tall, shoulders back, and extending your arms down through a clean range of motion.
There are only so many times that your vertabrae can take the extra demands of an overcompromised posture. Leave your ego in the parking lot and execute the lifts properly in order to ensure a decreased chance of injury.