The Push Up
Not just an arm & chest exercise...in fact, hardly even.
The Push Up, done correctly, should utilise the gluts; abs; obliques & deep core muscles for trunk stability. It should be a smooth, controlled movement.
The inability to perform a good, strong pushup could be a sign of trunk instability.
Here's an example of a regression for the traditional Push Up, done on the knees. If this is still too difficult to do with good form then a further regression is necessary and extra work is needed to improve core stability. A good further regression is using a fitball against the wall in order to strengthen the Push Up position.
Look out for a further post on the Fitball Push Up.