Get a grip
Something as seemingly insignificant as how to grip a bar can actually have a surprising effect on what can be gained from an exercise. Indeed, a poor grip can cause early onset fatigue and, in some cases, may cause injury. Equally, changing the grip can change the focus on which muscle is trained.
Avoid fatiguing the wrong muscles
For instance, functions such as Pullups, Seated Row and Bicep Curl put a lot of stress on smaller muscles in the forearm, often causing the set to be cut short due to their fatigue. By gripping the bar at the base of the fingers (where the fingers join the hands), it helps to gain a stronger grip without the stress on the forearm. Thus, ensuring the desired muscles are fully fatigued and hence maximising benefit from the exercise. To demonstrate the effectiveness of this grip, try gripping a finger from the opposite hand this way; you will notice a marked difference in the grip-strength without much effort.
Personally, I find even placement of the thumbs during a lift can have a significant affect. I prefer to remove the thumbs from the grip whenever safe to do so during pressing movements. This removes the strain on the forearms and tends to keep the wrist in line with them, thus reducing stress; something of particular benefit for heavy exercises such as Shoulder-Press or Bench-Press.
Focus on the right muscles
By varying the grip or changing the orientation of the wrist, you can alter which muscle is fatigued during a movement, helping you to focus on the right muscles.
A common example of where grip can alter the impact of an exercise is demonstrated by varying the grip during the Arm Curl; something the EZ (Easy) Bar was designed for. It is shown that, by gripping the bar with the wrist turned inward toward the body, it tends to work on the outer part of the Bicep more. Conversely, by gripping with the wrist turned outward the inner Bicep is affected to a greater extent.
Other examples include Tricep Press and pull-ups.
In summary, when performing an exercise, try to think about its goal, focus on the muscles you want to fatigue and adjust your grip accordingly.