Hm. Lots of factoids in the answers without a clear picture. Let me have a go at it.
Bias: This is required to put the transistors into their operating region. On top of the bias we apply a signal. Without the bias, the transistor cannot operate (at least not for linear amplifiers, class E and digital is a different story). For instance on a single transistor amplifier the gate bias voltage sets the operating point while the signal is modulated on top of the bias. (lets say 1V bias and around that you can apply a sine wave of 100mV amplitude which gets amplified).
Offset: Usually this refers to an unwanted property. Lets say an ideal comparator with two inputs changes state if one input is fixed and another input crosses the same voltage. In the presence of an offset voltage, the change in the output will not occur at V1=V2 but at V1=V2+Voff.
I believe you may have been reading the datasheet of an opamp with bipolar inputs. In this case they refer to the input currents caused by the base emitter diode as the input bias currents.
In a analog design biasing is required to keep the transistors in design in strong saturation
This is incorrect. I have designed lots of amplifiers with transistors in subthreshold and transistors for compensation in linear region. All of these are biased. For some reason people seem to mistake strong inversion with saturation a lot but that's a different issue.
bias is the external source that you are applying to op-amp for it to properly operate while the offset is actually used for you to set your differential gain to be zero
Wrong. The differential gain will always be the same, you set the output voltage to zero (or mid point for single supply amplifiers). Vout=Adiff(V1-V2+Voff). In that equation Adiff stays constant.
dalraist: Very good answer. I would like to add that offset usually consists of stochastic (random) offset and deterministic (systematic) offset. The latter can be eliminated with perfectly symmetrical designs and layouts, but believe me a lot of thought goes into this, especially for high current stuff. Random offset can be minimized (device size, layout) but not eliminated except by system design (autozero, chopper, CDS).
drDOC: I read your posts too late, I could have saved myself a lot of typing