Meaning of firing angle in the context of a DC input supply
I've learnt the meaning of firing angle only in the case when the input supply is AC. 




Figure 1. The upper trace shows the trigger delayed close to the end of the cycle. The resultant effective voltage is low. The lower trace shows the trigger close to the start of the cycle. This will result in close to full voltage. The relationship between phase angle delay and resultant RMS voltage is graphed on the right. Source: Opto-triacs, solid-state relays (SSR), zero-cross and how they work.

Your first image is for a triac-controlled variable phase-angle AC power control. There is no DC involved.


  So what does firing angle of 60,120,180 degrees mean when the input supply is DC? Like in a six step voltage source inverter ....


Your diagram shows an outline diagram of a circuit designed to generate a three-phase AC supply from a DC supply.


  Moreover, what is the meaning of 180 degree conduction mode? I get that the IGBT stays conducting for 180 degrees, but what 180 degrees of what cycle? What is the base/reference against which this 180 degrees is measured?


Somewhere there is a master counter going 0, 1, 2, 3, 4, 5, 0, 1, 2, ...


S1 is closed for count 0, 1 &amp; 2.
S2 is closed for count 1, 2 &amp; 3.
etc.


To generate the three phase supply (very crudely) the sequence is:

Table 1. Inverter output polarity.

Step  Angle   A    B    C
0       0º    +    -    +
1      60º    +    -    -     
2     120º    +    +    -
3     180º    -    +    -
4     240º    -    +    +
5     300º    -    -    +








simulate this circuit &ndash; Schematic created using CircuitLab

Figure 2. Resultant three-phase output.



Figure 3. A sinusoidal three-phase supply. Source: Wikipedia Three-phase.

The simple switching of the original schematic doesn't quite replicate mains three-phase!