Current draw follows the formula I = E/R
Where:
I = Current in Amps
E = Volts
R = Resistance in Ohms
Essentially, turn on your toaster and the resistance in the heater elements determine the current draw for the circuit for a given household voltage.
Adding additional appliances or lights on the same circuit just adds parallel resistance to the circuit.
The formula for parallel resistance is 1/RTotal = 1/R1 + 1/R2 + ... Rn
Simply said, if a toaster's resistance is 100 Ohms and you turn on a light that is 300 Ohms, the total circuit's resistance is now 75 Ohms.
Plugging in the total resistance to the original formula yields I = 117 / 75 = 1.56 Amps.
Now, as for the source, the limit on the current supplied is actually the power company and the wires to your building. Typically, that can be 100s of Amps, but you don't want that much current running through house wires, so you install breakers in the breaker box that trip when you exceed the limit for a given circuit.
So, it is not the power company that limits the current flowing through a given circuit, but the devices themselves and their impedance (similar to resistance) and the household voltage (i.e., 117 VAC or 220 VAC).
On last thing. For devices that use transformers like TVs, stereos, computers, etc., the correct term is impedance, not resistance. The reason is that impedance is the resistance at a specific frequency, whereas resistance is a term that is not frequency dependent. Transformers generally run at 50 Hz or 60 Hz (depending on where you live) and some do both, but their efficiency may suffer if you use the wrong frequency. I am probably getting off-track here, so I hope this helps answer your question.