A link-front coat or jacket is one that is closed by two connected buttons similar to a cufflink.
The original version appears to be early morning coats that used a stand-alone pair of linked buttons which were inserted into buttonholes on either side of the coat’s front. Over time it became more common for the buttons to be integrated into the jacket: one button was sewn to the outside of the jacket as normal while a second with a longer shank was sewn to the inside of the jacket behind the first. This configuration allowed the jacket to be buttoned by the usual method or by the linked method, the latter providing extra room for the inevitable post-dinner expansion of the wearer’s waistline.
Early dinner jackets were first worn open (and in fact were often constructed without buttons) but when they began to be worn closed in the 1910s it was often with a link button. A 1913 Vanity Fair article describing the greater latitude allowed in dinner clothes of the time mentions a particularly ornate version of the closure: “One is not confined to the one button single breasted garment. We shall see it with two buttons and also those which are caught with the jeweled link.”
Link-front dinner jackets pretty much disappeared from US fashion and etiquette guides by World War Two. However, they have remained a going concern in the UK and can still be obtained from a number of high-end menswear shops today.