If you pick any two points, the difference in gravitational potential between them tells you the amount of work that needs to be done against gravity to get from one point to the other, or the amount of gravitational energy released in moving between the two points in the other direction. It is usually measured in Joules of energy per kilogram of mass. For example, the top of a waterfall has a higher gravitational potential than the bottom, so every kilogram of water that falls gains that much kinetic energy on the way down, minus whatever is absorbed by drag - at Niagara Falls, that would be 500J per kilogram, energy that is ripe for harvesting as hydroelectric power.
Electrical potential is a close analogue in electromagnetism - the potential difference between two points on a circuit is what we are all familiar with as voltage. Again, this is a measure of the amount of energy that is gained or lost by a given amount of stuff moving from one point to another - a coulomb of electrons travelling from one pole of a 9V battery to the other will gain 9J, for example.
At sea level on Earth, a kilogram of matter that drops a metre will gain 9.8J of kinetic energy, because the local gravitational field strength is 9.8J/kg m, which is the same as saying that it is 9.8N/kg, or 9.8m/s2. The fact that the three units are equivalent might not be immediately obvious, but it is fairly easy to prove, and it brings out three different ways of looking at gravitational field strength, each with their uses. It can be thought of as a force that is proportional to mass; a constant rate of acceleration; or the rate of change of gravitational potential. Going upwards can hence be seen either as pushing against the force of gravity, or working your way towards the top of a potential 'gravity well'.
So far we have only talked about the difference in gravitational potential from one point to another. Sometimes it is useful to be able to talk about it in absolute terms, and for that, the convention is to compare the current location with a point so far from anything that it is completely unaffected by gravity. The gravitational potential in that case tells us how much energy it would take to escape from the local gravity well completely. To take an example, if it would take 1,000J of energy to fling a kilogram of matter out to the edge of space from here, then the gravitational potential right here is -1,000J/kg. That minus sign might seem like an oddity, but there is a good reason why gravitational potential energy is conventially considered to be negative; otherwise it would be almost impossible to define it absolutely. We need to define it with reference to something, and a point at infinity is really the only way that makes sense.
One curious consequence of defining potential (and potential energy) as negative is that it makes it possible for the total amount of energy in the universe to be exactly zero. This makes the creation of the universe - or indeed, a multiverse - surprisingly compatible with the conservation of energy.
The gravitational potential (or more accurately its relativistic equivalent) is also what determines the degree of gravitational time dilation according to Einstein's theory of General Relativity. The curvature of spacetime causes time to slow down around any massive body, and the amount of slowing depends on gravitational potential.