Do half link chains work on any bike and what is their purpose?

As mentioned in another answer, the main purpose of half link chains is to have more control over the chain length and hence the wheelbase of your bike (assuming you have rear facing or slideable drop-outs). A centimetre difference in wheelbase will change the feel of the bike quite a bit, especially on a short bike like a BMX.

Half link chains can only be used on single speed bikes* because their side profile won't match/mesh with the ramps that help lift it onto the next gear.

• Thinking about it they should also work on a bike with internal hub gears.

Half link chains also won't work with thick/thin chainrings as every link is thin on a half link chain.

Half link chains most commonly come in 1/8" width for BMX and general single speed use. You can also get them in 3/32" width for use with a mountain bike drivetrain, a lot of dirt jump mountain bikes run a mountain bike crankset with a 3/32" chainring for example.

Note: Modern mountain bike chains are actually a little bit narrower than 3/32" once you get to 9 speed or more. The difference is enough to stop a 3/32" chain working with 9+ speed gears but a 3/32" chain will still work with a 12 speed compatible chainring.

Using a 1/8" chain on a single speed mountain bike drivetrain might work but there would be a lot more side to side play which can't be good for wear! It also might not fit through a chain device designed for 3/32".