What is affecting the power supply current rating?
I'm assuming you're talking about a AC-DC power supply.

Ratings:

The main ratings on an AC supply that you can't usually exceed are:


Input voltage (and frequency): Don't give it higher than this voltage on the input.
Output current: Don't draw more than this amount of current from the output.


Exceeding either of those two will likely break something.

There are couple other basic ratings that a power supply has, but they relate to how the supply will behave if your adhere to the above:


Output voltage: At the specified input voltage, this will be the output voltage
Input current: At the maximum specified output current, this will be the maximum input current


Since your LED draws 4 amps, you are just fine connecting it to a 10 amp supply.

How they are determined:

Primarily by two things:


Heat and voltage the parts can take before breaking.
The supply behaving as specified.


Whenever you draw current through a wire, it heats up since some energy is lost. A transformer is two or more giant coils of wire wound around some kind of core, so the wires inside the transformer heat up with current. At some point, the temperature of the transformer will get hot enough that the insulation between the wires stops working as designed and your transformer shorts out between the windings. This causes the wiring ratio to change and the output voltage of the transformer to drop (or rise!!), often far enough so that the DC side of the supply stops working. The maximum current rating of a transformer is the current that can be drawn through it without heating it up to the point where its life is reduced. There are other factors as well in addition to winding resistance (for one, the core of the transformer has a maximum amount of magnetic flux it can contain before it stops behaving as expected), but heating is generally a good indicator if something is being pushed too hard.

For the input voltage rating, there is a point where the transformer will start shorting out because the input voltage is so high that it breaks down the insulation between the individual coils on the same winding or the insulation between the input (primary) and output (secondary) winding. This voltage is known as the "isolation rating" or "breakdown voltage" and most properly certified transformers will have a breakdown voltage of at least a thousand volts for a short period of time.  However, that doesn't mean that you could put several thousand volts on the transformer and expect it to work. The magnetic properties of the transformer see to that.

As for the DC part of the supply, on a basic level it consists of a diode, capacitor, and possibly some semiconductors. Each of these components also has a voltage and current rating that needs to be respected. The exact ratings are determined through a variety of ways, but again, they will heat up when this is exceeded. Putting too much voltage into the supply could cause too much voltage to be put into the DC components. Drawing too much current from the output could exceed the current ratings of these components. Certain components, such as capacitors, are rated to operate for a certain number of hours at some temperature (105C, 85C, etc). The output current rating of the supply is also calculated to prolong the life of these kinds of components so that they will hopefully last as long as the customer expects them to. This is also a place where corners are cut, since the power supply might get just warm enough so that the useful lifespan of the parts is shortened, but it will work good enough for some time and use cheaper parts.

In reality, perhaps you could draw more current through a supply than it has specified, or you could put more input voltage to it. However, the supply wasn't designed to work within those conditions and could misbehave. The specifications on the supply also reflect the designer's intentions for how the supply is to be used.