Single Speed/BMX (1/8”) chains are designed for non-derailleur applications only. 1/8” chains have less deflection or lateral flex than multi-speed chains. Multi-speed chains are optimized to flow through the various cogs and chainrings as the rider shifts gears. The light colored chain is 1/8" the darker chain is 3/32"
The Single Speed/non-derailleur chain has an internal width of 1/8” (where the teeth of the chainring nests inside see photo) which is wider than the 3/32” internal width of a derailleur equipped bicycle chain. As you might have surmised, the width of the chainring compliments the width of the chain so single speed chainrings are wider than multi-speed chainrings. While the widths of the chains differ the pitch (distance between the rollers) both chains measures ½”.
Most modern multi-speed chainrings are equipped with ramps and pins (see photo photo above) which are designed to facilitate smoother more accurate shifting. Single speed rings of course lack these features.
Bicycles with internally geared mechanisms (usually hubs) will often use the 1/8” chain since there is no derailleur on the bike and the chain remains fixed on one chainring and one rear cog.
Often you will see a 1/8” chain on a single speed bicycle with 3/32” cog and chainring. The wider chain on the skinnier rings poses no problems. Some feel the wider chain is stronger and they choose this set-up hoping for a more robust drive train. Others prefer the 3/32” chain and rings for single speed applications. These riders may spec their bikes in this way to save a bit of weight or to exploit the increased deflection of the 3/32” chain which may run quieter and smoother in set-ups where the chain line is not quite perfect.