'Centre of gravity' or 'centre of mass'
The centre of mass of an object is the point through which the weight of that object acts in a uniform gravitational field - that is why it is often called the centre of gravity.
The Physics syllabus says it is 'that point at which the mass of the body may be thought to be concentrated'. That is because we can draw the weight arrow (representing the weight of the whole object) from that point. However the weight isn't really concentrated there - there is just an even distribution of weight on either side of that point.
To stay upright we have to position our centre of gravity within our 'base' (our contact with the support we are standing on). Humans can get very good at balancing...
If you take moments about that point (along any plane) the sum of the moments will be zero - it will balance!
The centre of gravity point is always vertically below the point from which it is suspended from. Whatever orientation you suspend the object from this is always the case!
It is easy to find the centre of mass of a regular object. You just have to draw the lines of symmetry for it. Where they cross is the centre of mass. For a non-uniform object you have to carry out a little experiment to find the C of G.
Here is a mind boggling demo where the centre of gravity is outside the object...
... and here is the explanation of it!
Stability and Centre of Gravity
Remember - for an object to be stable its centre of gravity has to be below its point of suspension.