Consider this very basic scenario:

• I grab a rock, and cast the cantrip Light on it, choosing a violet hue as the color for the light.


• I set the rock down on the ground


• I cast Fog Cloud in the area surrounding the lit-up rock

Light reads as casting bright light in a 20' radius, and dim light another 20' away, and Fog Cloud (cast as a first level spell) has a 20' radius. So what happens in this scenario?

Does the

• 
  Fog Cloud completely obscure the light source, including the dim light shining 20' outside the Fog Cloud itself?


• 
  Light shine outside the Fog Cloud, providing dim light within 20' of the fog cloud?


• light from the Light spell diffuse through the Fog Cloud, creating a colored fog cloud?

First off, officially speaking, a spell only does what it says it does. A fog cloud spell creates a sphere of fog, which creates a heavily obscured area. It doesn't explicitly block or douse lights inside the area, as a darkness spell does, so the next thing is to look at what "obscured" does.

A heavily obscured area [...] blocks vision entirely. A creature in a heavily obscured area effectively suffers from the blinded condition

The errata adds:

A heavily obscured area doesn't blind you, but you are effectively blinded when you try to see something obscured by it.

Nothing in there mentions blocking light, only that you can't see through it. This fits well with our everyday understanding of what fog is like; light diffuses through it but is not actually blocked.

For the purpose of the game, then, the light (bright or dim) would extend beyond the fog cloud to the normal distance. The visual effect of this is up to the DM, but yes, in real life, a light in a fog cloud makes the cloud glow from inside, so it's reasonable for the DM to rule that the cloud lights up that way.

According to this tweet by Jeremy Crawford, Fog Cloud blocks vision, not light:

The fog cloud spell blocks vision. It doesn't eliminate light.

So the fog cloud will make it impossible to see your rock, but the rock will cast light inside or outside the fog cloud as normal.

The Light Source is completely obscured

In the Player's Handbook, in Chapter 8: Adventuring, under Environment section, the paragraph on Vision and Light reads as follows, with respect to "Heavy Obscurement":

A heavily obscured area—such as darkness, opaque fog, or dense foliage—blocks vision entirely. A creature in a heavily obscured area effectively suffers from the blinded condition (see the appendix). A heavily obscured area doesn't blind you, but you are effectively blinded when you try to see something obscured by it.

Since the Light source is inside the Fog Cloud, and the Fog Cloud specifically dictates that it generates a "Heavily Obscured" area, it would not be possible to see the Light source—or anything else—inside the Fog Cloud without some kind of ability to see through the Fog Cloud itself.