To analyze circuits containing diodes, it's essential to consider the diode's forward voltage drop, typically between 0.6 to 0.7 volts for silicon diodes. In a circuit without a resistor, the current is zero if the voltage is below this drop, while it can approach infinite current if the voltage exceeds it, risking damage to the diode. The diode equation provides a detailed analysis but may be unnecessary for basic calculations. Understanding the implications of connecting a diode directly to a voltage source without a resistor is crucial, as it can lead to excessive current and potential failure of the diode. Proper circuit design should always include resistors to limit current and protect components.