An ideal current source is able to change its potential difference at the output to keep the same current at the output regardless of the load imepedance there.
Some current sources, like the one you can make from a 3-terminal voltage regulator, operate like pjc50 says by using feedback control to adjust a voltage source until the output current is the desired value.
Others, like a transistor current mirror, inherently by design are high-impedance sources. Just like a resistor, they don't have to "adjust their voltage" to reflect changing currents, it's simply their nature that they will provide a nearly-constant current for widely different load voltages.
Here's an example of a current mirror circuit:
The output is sourced from the collector of Q2 and is controlled by the value of R2. Because the pnp BJT has inherently high output equivalent resistance, this will act as a current source for a wide range of output voltages.
Real current sources are able to do this till a limit, I think. I wish to know how do they do it?
The limit depends on the type of current source. A current mirror, for example, can be designed to either source or sink current, but not both, so that sets a limit on the compliance voltage at one end. At the other end, the limit is often set by the supply current and the load rather than the current mirror circuit itself.
In the example above, the current mirror will be able to source roughly equal current for voltages below about
+V - 0.2 V. The lower limit on output voltage will be determined by the power consumption capabilities of Q2.
Voltage sources have similar (complementary) limits. Typically a voltage source can only supply so much current, before it's output starts to drop below specification, or the source could even be damaged.
Also, what exactly does "load" mean in circuits,
When we are talking about sources, the load is whatever the source is supplying power to.
When a component loads another too much, does it source too much current? ... Does the concept of loading apply differently to voltage and current source?
For a voltage source, we might say the load is "too much" for the source when it requires too much current to maintain the desired voltage. This would be a low impedance load.
For a current source, we might say the load is too much for the source when it requires too much voltage to maintain the desired current. This would be a high impedance load.
So yes, what is an excessive load is different for voltage and current sources.