forward bias
Forward bias means connecting a PN junction the 'easy' way so that current can flow. You put the positive terminal of your source on the P side (the anode) and the negative terminal on the N side (the cathode). Do that with enough voltage and the diode wakes up and conducts, like opening a valve that lets water rush through.
Here is the mechanism in plain steps. The external voltage pushes holes from the P side and electrons from the N side toward the junction. As they pile in, the depletion region gets thinner and the built-in barrier shrinks. Once you exceed about 0.6 to 0.7 V in silicon, the barrier is low enough that carriers pour across and recombine, and a large current flows for only a little more voltage. Below that, only a trickle gets through.
In a real circuit you almost never set the diode's voltage directly — you set the current with something else (a resistor, a supply, a transistor) and let the diode settle at whatever forward voltage that current needs, usually near 0.7 V. That is why a diode in forward bias behaves like a roughly-fixed 0.7 V drop in series with your circuit, not like a resistor.
Put a silicon diode and a 1 kilohm resistor in series across a 5 V supply, diode pointing the way the current wants to go. About 0.7 V lands on the diode, leaving 4.3 V across the resistor, so the current is 4.3 V / 1 kilohm = 4.3 mA.
Forward-biased, the diode 'eats' ~0.7 V; the series resistor sets the current.
There is no single magic turn-on voltage. The 0.6-0.7 V figure is for ordinary silicon at modest currents; an LED needs ~1.8-3.3 V, a Schottky only ~0.2-0.4 V. 'Forward' just means biased to conduct, whatever that device's drop happens to be.