The number of speeds a bike has is a measure of the number of different gears the rider can choose from. Ignoring a couple of historic variations...
When we talk about chain speeds, we count the number of separate sprockets, or gears, that make up the cassette mounted on the rear wheel. A 10 speed chain is used on a bike with 10 sprockets in the cassette and an 11 speed chain is used on one with 11 sprockets in the cassette. Etc.
If you have numbers on the shifter, this will be the same as the highest number on the right hand shifter. Otherwise it’s a case of counting.
The chains themselves get narrower (on the outside dimension) as the number of speeds increases, because the sprockets are squeezed into very nearly the same space, so are placed closer together and the chain needs to fit between them.
The length, the weight, the chain connector and locking mechanism are all separate design choices, but it is considered essential to match the number of speeds on the bike to the speeds of the chain. The user in this question was having problems with a 10 speed chain on a 9 speed bike, falling between the chainrings.
As for the historic exceptions, these would include referencing ‘vintage’ bikes and/or bikes where the number of sprockets at the back is less than 8.