I'm watching a launch sit on hold because the sky is overcast. The announced hold reason was "thick cloud cover, anvil cloud". So they have a checklist. What is on that checklist? How do they assess that checklist - for instance, how far away does an anvil cloud have to be for them to consider conditions a go? In this launch, they are aiming for a gap in the conditions that they are calculating to arrive at a particular time, and adjusting the countdown accordingly. They must be looking at the radar and judging when the cloud cover and surrounding clouds will fall within acceptable parameters.
As discussed in the related questions TildalWave posted in a comment it really depends on the rocket (because I have it here's an article about a Soyuz launch in a snowstorm) and what the range they're launching from supports.
Considering you asked on April 27th you were probably watching the SpaceX TurkmenÄlem52E/MonacoSat (Thales) launch. For that specific launch I found this screenshot of the launch commit criteria: That's sourced from this reddit post. I don't know where he got the screenshot but it's an interesting post all around.
Note that the checklist breaks it down into separate groups for the range and the Falcon 9, this wikipedia page provides more detail but doesn't give any such separation. On the other hand, the wikipedia page is sourced from this NASA document which only includes the Liftoff Winds rule from the Falcon 9 specific criteria and otherwise appears to match the range criteria. Also, if I remember correctly the first stage landing criteria was a nice to have, not a requirement to launch (maybe why it's marked as N/A).
To quote wikipedia (since it's better formatted for copying text out of than the NASA pdf), the Falcon 9 cannot be launched under the following conditions:
- sustained wind at the 162 feet (49 m) foot level of the launch pad in excess of 30 knots (56 km/h; 35 mph)
- upper-level conditions containing wind shear[quantify] that could lead to control problems for the launch vehicle.
- launch through a cloud layer greater than 4,500 feet (1,400 m) thick that extends into freezing temperatures
- launch within 19 kilometres (10 nmi) of cumulus clouds with tops that extend into freezing temperatures,
- within 19 kilometres (10 nmi) of the edge of a thunderstorm that is producing lightning within 30 minutes after the last lightning is observed.
- within 19 kilometres (10 nmi) of an attached thunderstorm anvil cloud
- within 9.3 kilometres (5 nmi) of disturbed weather clouds that extend into freezing temperatures
- within 5.6 kilometres (3 nmi) of a thunderstorm debris cloud,
- through cumulus clouds formed as the result of or directly attached to a smoke plume,
The following should delay launch:
- delay launch for 15 minutes if field mill instrument readings within 9.3 kilometres (5 nmi) of the launch pad exceed +/- 1,500 volts per meter, or +/- 1,000 volts per meter
- delay launch for 30 minutes after lightning is observed within 10 nautical miles (19 km; 12 mi) of the launch pad or the flight path
This is a hard question to answer generally (without just resorting to "it depends), but if you're interested in other rockets the wikipedia link above also includes criteria for Atlas V and the Space Shuttle.